TEAM PROJECT. Series for Dirichlet and Nemnann (a) Show

This happens if a PDE involves derivatives with respect to one variable only (or can be transformed to such a form), so that the other variable(s) can be treated as parameter(s). Solve for u = u(x. y):

This happens if a PDE involves derivatives with respect to one variable only (or can be transformed to such a form), so that the other variable(s) can be treated as parameter(s). Solve for u = u(x. y):

This happens if a PDE involves derivatives with respect to one variable only (or can be transformed to such a form), so that the other variable(s) can be treated as parameter(s). Solve for u = u(x. y):

This happens if a PDE involves derivatives with respect to one variable only (or can be transformed to such a form), so that the other variable(s) can be treated as parameter(s). Solve for u = u(x. y):

This happens if a PDE involves derivatives with respect to one variable only (or can be transformed to such a form), so that the other variable(s) can be treated as parameter(s). Solve for u = u(x. y):

This happens if a PDE involves derivatives with respect to one variable only (or can be transformed to such a form), so that the other variable(s) can be treated as parameter(s). Solve for u = u(x. y):

This happens if a PDE involves derivatives with respect to one variable only (or can be transformed to such a form), so that the other variable(s) can be treated as parameter(s). Solve for u = u(x. y):

This happens if a PDE involves derivatives with respect to one variable only (or can be transformed to such a form), so that the other variable(s) can be treated as parameter(s). Solve for u = u(x. y):

This happens if a PDE involves derivatives with respect to one variable only (or can be transformed to such a form), so that the other variable(s) can be treated as parameter(s). Solve for u = u(x. y):

This happens if a PDE involves derivatives with respect to one variable only (or can be transformed to such a form), so that the other variable(s) can be treated as parameter(s). Solve for u = u(x. y):

This happens if a PDE involves derivatives with respect to one variable only (or can be transformed to such a form), so that the other variable(s) can be treated as parameter(s). Solve for u = u(x. y):

This happens if a PDE involves derivatives with respect to one variable only (or can be transformed to such a form), so that the other variable(s) can be treated as parameter(s). Solve for u = u(x. y):

(Fundamental Theorem) Prove Fundamental Theorem I for second-order PDEs in two and three independent variables.

Verify (by substitution) that the given function is a solution of the indicated PDE. Sketch or graph the solution as a surface in space.

Verify (by substitution) that the given function is a solution of the indicated PDE. Sketch or graph the solution as a surface in space.

Verify (by substitution) that the given function is a solution of the indicated PDE. Sketch or graph the solution as a surface in space.

Verify (by substitution) that the given function is a solution of the indicated PDE. Sketch or graph the solution as a surface in space.

Verify (by substitution) that the given function is a solution of the indicated PDE. Sketch or graph the solution as a surface in space.

Verify (by substitution) that the given function is a solution of the indicated PDE. Sketch or graph the solution as a surface in space.

Verify (by substitution) that the given function is a solution of the indicated PDE. Sketch or graph the solution as a surface in space.

Verify (by substitution) that the given function is a solution of the indicated PDE. Sketch or graph the solution as a surface in space.

Verify (by substitution) that the given function is a solution of the indicated PDE. Sketch or graph the solution as a surface in space.

Verify (by substitution) that the given function is a solution of the indicated PDE. Sketch or graph the solution as a surface in space.

Verify (by substitution) that the given function is a solution of the indicated PDE. Sketch or graph the solution as a surface in space.

Verify (by substitution) that the given function is a solution of the indicated PDE. Sketch or graph the solution as a surface in space.

TEAM PROJECT. Verification of Solutions (a) Wave equation. Verify that u(.\". t) = vex + ell + w(x - el) with any twice differentiable functions v and w satisfies (I). (b) Poisson equation. Verify that each u satisfies (4) with f(x. y) as indicated. u = X4 + y4 f = 12(x 2 + y2) U = cos.\ sin y f = -2 cos x sin y u = .vlx f = 2)'lx3 (c) Laplace equation. Verify that u = IIV x 2 + y2 + Z2 satisfies (6) and u = In (x2 + y2) satisfies (3). Is u = l/V x2 + y2 a solution of (3)? Of what Poisson equation? (d) Verify that II with any (sufficiently often differentiable) v and w satisfies the given PDE. 1I = vex) + II"(Y) !I = v(xhl"(y) 1/ = vex + 31) + w( \" - 31)

(Boundary value problem) Verify that the function u(x, y) = a In (x 2 + ,.2) + b satisfies Laplace's equation (3) and determine a and b so that u satisfies the boundary conditions 1I = 110 on the circle x 2 + .1'2 = 1 and II = 0 on the circle.\"2 + y2 = 100.

Solve!Ix = 0, lly = 0

Solve U"X = 0, !lxy = 0

Solve l/xx = n, llyy = n

In the vertical direction we have two forces. namely, the vertical components - T] sin ll' and T2 sin f3 of Tl and T2; here the minus sign appears because the component at P is directed downward. By Newton's second law the resultant of these two forces is equal to the mass p ~x of the portion times the acceleration a2 uICJ(2, evaluated at some point between x and x + ~x; here p is the mass of the un deflected string per unit length. and ~x is the length of the portion of the undeflected string. (~ is generally used to denote small quantities; this has nothing to do with the Laplacian V2, which is sometimes also denoted by ~.) Hence

Using 0), we can divide this by T2 cos f3 = Tl cos ll' = T, obtaining (2) Tl sin ll' p ~x a2 u = tan f3 - tan ll' = -T 0"(2 Tl cos ll' Now tan ll' and tan f3 are the slopes of the string at x and x + ~x: tan ll' = (~u) I dx x and tanf3= (- . au) I ax x+.lx Here we have to write partial derivatives because u depends also on time (. Dividing (2) b) ~x, we thus have If we let ~x approach zero, we obtain the linear PDE

f the second order. The physical constant Tip is denoted by c 2 (instead of c) to indicate 540 CHAP. 12 Partial Differential Equations (PDEs) that this constant is positive. a fact that will be essential to the form of the solutions. "One-dimensional" means that the equation involves only one space variable. x. In the next section we shall complete setting up the model and then show how to solve it by a general method that is probably the most imp0l1ani one for PDEs in engineering mathematics.

Find u(x, t) for the string of length L = 1 and c2 = 1 when the initial velocity is zero and the initial deflection with small k (say, 0.01) is as follows. Sketch or graph U(X, 1) as in Fig. 288.

Find u(x, t) for the string of length L = 1 and c2 = 1 when the initial velocity is zero and the initial deflection with small k (say, 0.01) is as follows. Sketch or graph U(X, 1) as in Fig. 288.

Find u(x, t) for the string of length L = 1 and c2 = 1 when the initial velocity is zero and the initial deflection with small k (say, 0.01) is as follows. Sketch or graph U(X, 1) as in Fig. 288.

Find u(x, t) for the string of length L = 1 and c2 = 1 when the initial velocity is zero and the initial deflection with small k (say, 0.01) is as follows. Sketch or graph U(X, 1) as in Fig. 288.

Find u(x, t) for the string of length L = 1 and c2 = 1 when the initial velocity is zero and the initial deflection with small k (say, 0.01) is as follows. Sketch or graph U(X, 1) as in Fig. 288.

Find u(x, t) for the string of length L = 1 and c2 = 1 when the initial velocity is zero and the initial deflection with small k (say, 0.01) is as follows. Sketch or graph U(X, 1) as in Fig. 288.

Find u(x, t) for the string of length L = 1 and c2 = 1 when the initial velocity is zero and the initial deflection with small k (say, 0.01) is as follows. Sketch or graph U(X, 1) as in Fig. 288.

Find u(x, t) for the string of length L = 1 and c2 = 1 when the initial velocity is zero and the initial deflection with small k (say, 0.01) is as follows. Sketch or graph U(X, 1) as in Fig. 288.

Find u(x, t) for the string of length L = 1 and c2 = 1 when the initial velocity is zero and the initial deflection with small k (say, 0.01) is as follows. Sketch or graph U(X, 1) as in Fig. 288.

Find u(x, t) for the string of length L = 1 and c2 = 1 when the initial velocity is zero and the initial deflection with small k (say, 0.01) is as follows. Sketch or graph U(X, 1) as in Fig. 288.

(Frequency) How does the frequency of the fundamental mode of the vibrating string depend on the length of the string? On the mass per unit length? What happens to the string if we double the tension? Why is a contrabass larger than a violin?

(Nonzero initial velocity) Find the deflection II(X, t) of the string of length L = 7f and ("2 = I for zero initial displacement and "triangular" initial velocity IIt(x, 0) = 0.0 I x if 0 :;;; x ~ ~7f, IIt (X, 0) = 0.0 I (7f - x) if ~7f ~ x :;;; 7f. (Initial conditions with lIix, 0) ~ 0 are hard to realize experimentally.)

CAS PROJECT. Graphing Normal Modes. Write a program for graphing lin with L = 7f and c 2 of your choice similarly as in Fig. 284. Apply the program to 112, U3, 114' Also graph these solutions as surfaces over thext-plane. Explain the connection between these two kinds of graphs.

String. Show the following. (a) Substitution of x Il7fX (17) II(X, t) = L GnU) sin L n=l (L = length of the string) into the wave equation (I) governing free vibrations leads to [see (lO'~)J (18) 2_ Gn + An G - 0, (b) Forced vibrations of the string under an external force P(x, t) per unit length acting normal to the string are governed by the PDE (19) (c) For a sinusoidal force P = Ap sin wt we obtain (20) p ex n7fX = A sin wt = L knCt) sin L ' p n~l _ {(4A11l7f) sin wt k,,(t) - o (11 odd) (n even). Substituting (17) and (20) into (19) gives 2 _ 2A ,,' Gn + An Gn - - (l - cos 1l7f) sm wt. f (d) (Resonance) Show that if An = w, then A - -- (1 - cos 1l7f)T cos WT. n7fW (e) (Reduction of boundary conditions) Show that a problem (1)-(3) with more complicated boundary conditions 11(0, t) = 0, u(L, t) = h(t), can be reduced to a problem for a new function v satisfying conditions v(O, t) = v(L, f) = 0, vex. 0) = fl(x), Vt(x, 0) = gl(X) but a nonhomogeneous wave equation. Him: Set II = V + I\" and determine w suitably.

By the prinCiples used in modeling the string it can be shown that small free vertical vibrations of a uniform elastic beam (Fig. 289) are modeled by the fourth-order PDE (21) (Ref. [CII]) where c 2 = EIIpA (E = Young's modulus of elasticity, I = moment of intertia of the cross section with respect to the y-axis in the figure, p = density, A = cross-sectional area). (Bending of a beam under a load is discussed in Sec. 3.3.)

By the prinCiples used in modeling the string it can be shown that small free vertical vibrations of a uniform elastic beam (Fig. 289) are modeled by the fourth-order PDE (21) (Ref. [CII]) where c 2 = EIIpA (E = Young's modulus of elasticity, I = moment of intertia of the cross section with respect to the y-axis in the figure, p = density, A = cross-sectional area). (Bending of a beam under a load is discussed in Sec. 3.3.)

By the prinCiples used in modeling the string it can be shown that small free vertical vibrations of a uniform elastic beam (Fig. 289) are modeled by the fourth-order PDE (21) (Ref. [CII]) where c 2 = EIIpA (E = Young's modulus of elasticity, I = moment of intertia of the cross section with respect to the y-axis in the figure, p = density, A = cross-sectional area). (Bending of a beam under a load is discussed in Sec. 3.3.)

By the prinCiples used in modeling the string it can be shown that small free vertical vibrations of a uniform elastic beam (Fig. 289) are modeled by the fourth-order PDE (21) (Ref. [CII]) where c 2 = EIIpA (E = Young's modulus of elasticity, I = moment of intertia of the cross section with respect to the y-axis in the figure, p = density, A = cross-sectional area). (Bending of a beam under a load is discussed in Sec. 3.3.)

By the prinCiples used in modeling the string it can be shown that small free vertical vibrations of a uniform elastic beam (Fig. 289) are modeled by the fourth-order PDE (21) (Ref. [CII]) where c 2 = EIIpA (E = Young's modulus of elasticity, I = moment of intertia of the cross section with respect to the y-axis in the figure, p = density, A = cross-sectional area). (Bending of a beam under a load is discussed in Sec. 3.3.)

By the prinCiples used in modeling the string it can be shown that small free vertical vibrations of a uniform elastic beam (Fig. 289) are modeled by the fourth-order PDE (21) (Ref. [CII]) where c 2 = EIIpA (E = Young's modulus of elasticity, I = moment of intertia of the cross section with respect to the y-axis in the figure, p = density, A = cross-sectional area). (Bending of a beam under a load is discussed in Sec. 3.3.)

Show that c is the speed of each of the two waves given by (4).

Show that because of the boundaty conditions (2). Sec. 12.3, the function f in (13) of this section must be odd and of period 2L.

If a steel wire 2 m in length weighs 0.9 nt (about 0.20 lb) and is stretched by a tensile force of 300 m (about 67.4 Ib). what is the corresponding speed of transverse waves?

What are the frequencies of the eigenfunctions in Prob.3?

Longitudinal Vibrations of an Elastic Bar or Rod. These vibrations in the direction of the x-axis are modeled by the wave equation Utt = r 2 uxx , c 2 = EI P (see Tolstov [C9]. p. 275). If the rod is fastened at one end. x = 0, and free at the other. x = L. we have 11(0. t) = 0 and II.AL, t) = O. Show that the motion corresponding to initial displacement u(x, 0) = f(x) and initial velocity zero is = II = 2: An sin Pnx cos Pnct, n=O 2 fL An = - f(x) sin PnX dx, L 0 Pn = (211 + 1)7T 2L

using (13), st...etch or graph a figure (similar to Fig. 288 in Sec. 12.3) of the deflection u(x. t) of a vibrating string (length L = L ends fixed, c = I) starting with initial velocity 0 and initial deflection (k small, say, k = 0.0 I).

using (13), st...etch or graph a figure (similar to Fig. 288 in Sec. 12.3) of the deflection u(x. t) of a vibrating string (length L = L ends fixed, c = I) starting with initial velocity 0 and initial deflection (k small, say, k = 0.0 I).

using (13), st...etch or graph a figure (similar to Fig. 288 in Sec. 12.3) of the deflection u(x. t) of a vibrating string (length L = L ends fixed, c = I) starting with initial velocity 0 and initial deflection (k small, say, k = 0.0 I).

using (13), st...etch or graph a figure (similar to Fig. 288 in Sec. 12.3) of the deflection u(x. t) of a vibrating string (length L = L ends fixed, c = I) starting with initial velocity 0 and initial deflection (k small, say, k = 0.0 I).

(Tricomi and Airy equations2) Show that the Tricomi equatioll YUxx + lIyy = 0 is of mixed type. Obtain the Airy equation G" - yG = 0 from the Tricomi equation by separation. (For solutions, see p. 446 of Ref. [GR I] listed in App. I.)

Find the type, transform to normal form. and solve. (Show the details of your work.)

Find the type, transform to normal form. and solve. (Show the details of your work.)

Find the type, transform to normal form. and solve. (Show the details of your work.)

Find the type, transform to normal form. and solve. (Show the details of your work.)

Find the type, transform to normal form. and solve. (Show the details of your work.)

Find the type, transform to normal form. and solve. (Show the details of your work.)

Find the type, transform to normal form. and solve. (Show the details of your work.)

Find the type, transform to normal form. and solve. (Show the details of your work.)

Find the type, transform to normal form. and solve. (Show the details of your work.)

Find the type, transform to normal form. and solve. (Show the details of your work.)

WRITING PROJECT. Wave and Heat Equations.Compare the two PDEs with respect to type, general behavior of eigenfunctions. and kind of boundary and initial conditions and resulting practical problems. Also discuss the difference between Figs. 288 in Sec. 12.3 and 292.

(Eigenfunctions) Sketch (or graph) and compare the first three eigenfunctions (8) with Bn = 1, c = I. L = 7T for t = 0, 0.2, 0.4, 0.6, 0.8. 1.0.

(Decay) How does the rate of decay of (8) with fIxed /J depend on the specific heat, the density, and the thermal conductivity of the material?

If the first eigenfunction (8) of the bar decreases to half its value within 10 sec, what is the value of the diffusi vity?

A laterally insulated bar of length 10 cm and constant cross-sectional area I cm2, of density 10.6 gmlcm3 , thermal conductivity 1.04 call( cm sec DC), and specific heat 0.056 cal/(gm DC) (this corresponds to silver, a good heat conductor) has initial temperature f(x) and is kept at ODC at the ends x = 0 and x = 10. Find the temperature u(x, t) at later times. Here, f(x) equals:

A laterally insulated bar of length 10 cm and constant cross-sectional area I cm2, of density 10.6 gmlcm3 , thermal conductivity 1.04 call( cm sec DC), and specific heat 0.056 cal/(gm DC) (this corresponds to silver, a good heat conductor) has initial temperature f(x) and is kept at ODC at the ends x = 0 and x = 10. Find the temperature u(x, t) at later times. Here, f(x) equals:

A laterally insulated bar of length 10 cm and constant cross-sectional area I cm2, of density 10.6 gmlcm3 , thermal conductivity 1.04 call( cm sec DC), and specific heat 0.056 cal/(gm DC) (this corresponds to silver, a good heat conductor) has initial temperature f(x) and is kept at ODC at the ends x = 0 and x = 10. Find the temperature u(x, t) at later times. Here, f(x) equals:

A laterally insulated bar of length 10 cm and constant cross-sectional area I cm2, of density 10.6 gmlcm3 , thermal conductivity 1.04 call( cm sec DC), and specific heat 0.056 cal/(gm DC) (this corresponds to silver, a good heat conductor) has initial temperature f(x) and is kept at ODC at the ends x = 0 and x = 10. Find the temperature u(x, t) at later times. Here, f(x) equals:

A laterally insulated bar of length 10 cm and constant cross-sectional area I cm2, of density 10.6 gmlcm3 , thermal conductivity 1.04 call( cm sec DC), and specific heat 0.056 cal/(gm DC) (this corresponds to silver, a good heat conductor) has initial temperature f(x) and is kept at ODC at the ends x = 0 and x = 10. Find the temperature u(x, t) at later times. Here, f(x) equals:

(Arbitrar~ temperatures at ends) If the ends x = 0 and x = L of the bar in the text are kept at constant temperatures VI and V2, respectively, what is the temperature UI(X) in the bar after a long time (theoretically, as t -'? "YO)? First guess, then calculate.

Tn Prob. 10 find the temperature at any time.

(Changing end temperatures) Assume that the ends of the bar in Probs. 5-9 have been kept at 100DC for a long time. Then at some instant. call it t = 0, the temperature at x = L is suddenly changed to ODC and kept at ODC, whereas the temperature at x = 0 is kept at 100De. Find the temperature in the middle of the bar at t = L 2. 3, 10, 50 sec. First guess, then calculate.

BAR UNDER ADIABATIC CONDITIONS "Adiabatic" means no heat exchange with the neighborhood. because the bar is completely insulated, also at the ends. PhysicolTnformation: The heat flux at the ends is proportional to the value of aulax there. 13. Show that for the completely insulated bar. ux(O, t) = 0, uAL, t) = 0, /leX, t) = f(x) and separation of variables gives the following solution, with An given by (2) in Sec. 11.3. "" u(x. t) = Ao + L An cos ";x e-{cn7T/L)2t

Find the temperature in Prob. 13 with L 11. C = 1, andf(x) = t"

Find the temperature in Prob. 13 with L 11. C = 1, andf(x) = 1

Find the temperature in Prob. 13 with L 11. C = 1, andf(x) = 0.5 cos 4x

Find the temperature in Prob. 13 with L 11. C = 1, andf(x) = 112 - x 2

Find the temperature in Prob. 13 with L 11. C = 1, and f(x) = ~11 - Ix - ~1I1

Find the temperature in Prob. 13 with L 11. C = 1, and f(x) = (x - ~11)2

Find the temperature of the bar in Prob. 13 if the left end is kept at ODC, the right end is insulated, and the initial temperature is Vo = const.

The boundary condition of heat transfer (19) -ux (1I, t) = k[II('IT. t) - uo] applies when a bar of length 'IT with c = I is laterally insulated, the left end x = 0 is kept at OC, and at the right end heat is flowing into air of constant temperature uo. Let k = I for simplicity, and 110 = O. Show that a solution is lI(x, t) = sin px e-p2t where P is a solution of tan p1I = - p. Show graphically that this equation has infinitely many positive solutions PI> P2, P3, ... , where Pn > 11 - i and lim (Pn - 11 + ~) = O. (Formula (19) is also known n_cc as radiation boundary condition, but this is misleading; see Ref. [C3], p. 19.)

(Discontinuous f) Solve (\), (2), (3) with L = 11 and f(x) = Vo = const (=1= 0) if 0 < x < 11/2, f(x) = 0 if 1112 < x < 11.

(Heat flux) The heat fiLix of a solution II(X, t) across x = 0 is defined by cp(t) = - KlIx(O, t). Find CPU) for the solution (9). Explain the name. Is it physically understandable that cp goes to 0 as t -'? x?

(Bar with heat generation) If heat is generated at a constant rate throughout a bar of length L = 'IT with initial temperature f(x) and the ends at x = 0 and 'IT are kept at temperature 0, the heat equation is Ut = c 2 uxx + H with constant H > O. Solve this problem. Hint. Set u = v - Hx(x - 'IT)/(2c 2 ).

(Convection) If heat in the bar in the text is free to flow through an end into the surrounding medium kept at OC, thePDEbecomesvt = c2 vxx - f3v. Show that it can be reduced to the form (I) by setting vex, t) = II(X, t)w(t).

Consider vt = c 2 vxx - v (0 < x < L, t > 0), v(O, t) = 0, veL, t) = 0, vex, 0) = f(x), where the term -v models heat transfer to the surrounding medium kept at temperature O. Reduce this PDE by setting vex, t) = u(x, t)w(t) with w such that U is given by (9), (10).

(Nonhomogeneous heat equation) Show that the problem modeled by and (2), (3) can be reduced to a problem for the homogeneous heat equation by setting u(x, t) = vex, t) + w(x) and determining w so that v satisfies the homogeneous PDE and the conditions v(O, t) = veL, t) = 0, v(x.O) = f(x) - w(x). (The term Ne- ax may represent heat loss due to radioactive decay in the bar.)

(Laplace equation) Find the potential in the rectangle o ~ x ~ 20, 0 ~ y ~ 40 whose upper side is kept at potential 220 V and whose other sides are grounded.

Find the potential in the square 0 ~ x ~ 2, 0 ~ y ~ 2 if the upper side is kept at the potential sin 47TX and the other sides are grounded.

CAS PROJECT. Isotherms. Find the steady-state solutions (temperatures) in the square plate in Fig. 294 with a = 2 satisfying the following boundary conditions. Graph isotherms. (a) II = sin TTX on the upper side. 0 on the others. (b) u = 0 on the vertical sides. assuming that the other sides are perfectly insulated. (c) Boundary conditions of your choice (such that the solution is not identically zero). YI a~ -, a x Fig. 294. Square plate

(Heat flow in a plate) The faces of the thin square plate in Fig. 294 with side a = 24 are perfectly insulated. The upper side is kept at 20C and tl1e other sides are kept at OC. Find the steady-state temperature II(X, y) in the plate.

Find the steady-state temperature in the plate in Prob. 31 if the lower side is kept at UO dc, the upper side at UI dc, and the other sides are kept at Oc. Hint: Split into two problems in which the boundary temperature is 0 on three sides for each problem.

Find the steady- ~tate temperature in the plate in Prob. 31 with the upper and lower sides perfectly insulated. the left side kept at Oc. and the right side kept at f(y)C.

(Radiation) Find steady-state temperatures in the rectangle in Fig. 293 with the upper and left sides perfectly insulated and the right side radiating into a medium at OC according to 1I,.(a. y) + hu(a, y) = O. " > 0 constant. (You will get many solutions since no condition on the lower side is given.)

Find formulas similar to (\7). (18) for the temperature in the rectangle R of the text when the lower side of R is kept at temperature f(x) and the other side~ are kept at Oc.

Using (6). obtain the solution of (I) in integral form satisfying the initial condition u(x. m = lex). wherelex) l if Ixl < a and 0 otherwise

Using (6). obtain the solution of (I) in integral form satisfying the initial condition u(x. m = lex). where f(x) = e- klxl (/.. > m

Using (6). obtain the solution of (I) in integral form satisfying the initial condition u(x. m = lex). where

Using (6). obtain the solution of (I) in integral form satisfying the initial condition u(x. m = lex). where

Using (6). obtain the solution of (I) in integral form satisfying the initial condition u(x. m = lex). where

Using (6). obtain the solution of (I) in integral form satisfying the initial condition u(x. m = lex). where

Using (6). obtain the solution of (I) in integral form satisfying the initial condition u(x. m = lex). where

Verify that II in Prob. 5 satisfies the initial condition.

CAS PROJECT. Heat Flow. (a) Graph the basic Fig. 296. (b) In (a) apply animation to "see" the heat flow in terms of the dccrease of temperature. (c) Graph u(x, t) with c = I as a surface over the upper xt-half-plane.

CAS PROJECT. Error Function (21) 2 rX erfx = -- J e- w2 dw ~ 0 This function is imp0l1ant in applied mathematics and physics (probability theory and statistics. thermodynamics. etc.) and fits our present discussion. Regarding it as a typical case of a special function defined by an integral that cannot be evaluated as in elementary calculus. do the following. (a) Sketch or gmph the bell-shaped curve [the curve of the integrand in (21 )J. Show that erf x is odd. Show that I b ~ e-w2 dw = 2 (erfb - erfa), a b I e- w2 dw = .y:;;: erf b. -b (b) Obtain the Maclaurin series of erf x from that of the integrand. Use that series to compute a table of erfx for x = OCO.OI)3 (meaning x = O. O.OL 0.02, .... 3). (c) Obtain the values required in (b) by an integration command of your CAS. Compare accuracy. (d) [t can be shown that erf (x) = 1. Confirm this experimentally by computing erf x for large x.(e) Let J(x) = 1 when x> 0 and 0 when x < O. Using erf(X) = 1, show that (12) then gives 1 x 2 u(x, t) = I I e- Z dz v 7f -x/(2cVtJ (t> 0). 1 IX 2 (g) Show that (t) = =-- e-s /2 ds "27f -ex: 569 (1) Express the temperature (13) in tenns of the error function. Here. the integral is the definition of the "distribution function of the normal distribution" to be discussed in Sec. 24.8.

T ily (sin f3 - sin a) = T .ly (tan f3 - tan a) = T~)' [ux(x + ~X, )'1) - ux(X, )'2)] where subscripts x denote partial derivative~ and Yl and .\'2 are values between), and )' + .ly. Similarly. the resultant of the vertical components of the forces acting on the other two sides of the portion is

Newton's Second Law Gives the POE of the Model. By Newton's second law (see Sec. 2.4) the sum of the forces given by (I) and (2) is equal to the mass p~A of that small SEC. 12.8 Rectangular Membrane. Double Fourier Series 571 portion times the acceleration a2 u/at2 ; here p is the mass of the undeflected membrane per unit area, and .lA = Llx Lly is the area of that portion when it is undeflected. Thus a2 u p Llx~)' at2 = T Lly [ux(x + .lX .'"1) - lIx(X, )'2)] + T ~.\ [lIiXI' )' + .1y) - Uy (X2, y)] where the derivative on the left is evaluated at some suitable point (x, Y) corresponding to that portion. Division by p .lx LlY gives a 2 u = T [lIx (X + .lx. )'1) - II"J', }'2) + lIy(x., y + ~)') - lIiX2. Y) ] .

If we let Llx and Lly approach zero, we obtain the PDE of the model (3) p This PDE is called the two-dimensional wave equation. The expression in parentheses is the Laplacian V2 u of u (Sec. 10.8). Hence (3) can be written (3') Solutions of the wave equation (3) will be obtained and discussed in the next section.

(Frequency) How does the frequency of the eigenfunctions of the rectangular membrane change if (a) we double the tension, (b) we take a membrane of half the mass of the original one, (c) we double the sides of the membrane? (Give reason.)

Determine and sketch the nodal lines of the eigenfunctions of the square membrane for m = I, 2, 3, 4 and n = I, 2, 3, 4.

Double Fourier Series. Represent f(x, y) by a series (15), where 0 < x < I. 0 < Y < I.

Double Fourier Series. Represent f(x, y) by a series (15), where 0 < x < I. 0 < Y < I.

Double Fourier Series. Represent f(x, y) by a series (15), where 0 < x < I. 0 < Y < I.

Double Fourier Series. Represent f(x, y) by a series (15), where 0 < x < I. 0 < Y < I.

Double Fourier Series. Represent f(x, y) by a series (15), where 0 < x < I. 0 < Y < I.

Double Fourier Series. Represent f(x, y) by a series (15), where 0 < x < I. 0 < Y < I.

CAS PROJECT. Double Fourier Series. (a) Wlite a program that gives and graphs partial sums of (\5). Apply it to Probs. 4 and 5. Do the graphs show that those partial sums satisfy the boundary condition (3a)? Explain Why. Why is the convergence rapid? (b) Do the tasks in (a) for Prob. 3. Graph a portion, say, 0 < x < ~, 0 < Y < ~, of several partial sums on common axes, so that you can see how they differ. (See Fig. 303.) (c) Do the tasks in (b) for functions of your choice.

CAS EXPERIMENT. Quadruples of F mn- Write a program that gives you four numerically equal "mn in Example I, so that four different Fmn correspond to it. Sketch the nodal lines of F 18, F 81, F 47, F74 in Example I and similarly for further F mn that you will find.

Deflection. Find the deflection u(x, y, t) of the square membrane of side 7r and c2 = 1 if the initial velocity is 0 and the initial deflection is k sin 2x sin 5y

Deflection. Find the deflection u(x, y, t) of the square membrane of side 7r and c2 = 1 if the initial velocity is 0 and the initial deflection is 0.1 sin x siny

Deflection. Find the deflection u(x, y, t) of the square membrane of side 7r and c2 = 1 if the initial velocity is 0 and the initial deflection is O.lxy( 7r - x)( 7r - y)

VerifY the discussion of the terms of (21) in Example 2.

Repeat the task of Prob. 2 when a = 4 and b = 1.

Verify the calculation of Bmn in Example 2 by integration by parts.

Find eigenvalues of the rectangular membrane of sides a = 2 and b = I to which there correspond two or more different (independent) eigenfunctions.

(Minimum property) Show that among all rectangular membranes of the same area A = ab and the same c the square membrane is that for which Un [see (10)] has the lowest frequency.

Double Fourier Series. Represent f(x, y) (0 < x < a, 0 < Y < b) by a double Fourier series (15).

Double Fourier Series. Represent f(x, y) (0 < x < a, 0 < Y < b) by a double Fourier series (15).

Double Fourier Series. Represent f(x, y) (0 < x < a, 0 < Y < b) by a double Fourier series (15).

Double Fourier Series. Represent f(x, y) (0 < x < a, 0 < Y < b) by a double Fourier series (15).

Deflection) Find the deflection of the membrane of sides a and b with c2 = I for the initial deflection f . 3'7TX . 4'7TY d" . I I . 0 (x, y) = sm -- sm -- an mltla ve oClty .

Repeat the task in Prob. 23 with c 2 = 1, for f(x, y) as in Prob. 22 and initial velocity O.

(Forced vibrations) Show that forced vibrations of a membrane are modeled by the PDE Utt = C2V2 U + PIp, where P(x, y, t) is the external force per unit area acting perpendicular to the xy-plane.

Why did we use polar coordinates in this section?

Work out the details of the calculation leading to the Laplacian in polar coordinates.

If l/ is independent of e, then (5) reduces to y 2 u = II'T + uTlr. Derive this directly from the Laplacian in Cartesian coordinates. 1

An alternative form of (5) is v2u = -r ill' iJ211 ( r-iJU) ilr + ae2 . Derive this from (5).

(Radial solution) Show that the only solution of y 2 u = 0 depending only on r = V.~ + i is u = a In r + b with constant a and b

TEAM PROJECT. Series for Dirichlet and Nemnann Problems (a) Show that lin = 1'71 cos lie. "n = rn sin ne, II = 0, I, ... , are solutions of Laplace's equation -V2 u = 0 with ,211 given by (5). (What would Un be in Cartesian coordinates'? Experiment with small II.) (b) Dirichlet problem (See Sec. 12.5) Assuming that term wise differentiation is permissible. show that a solution of the Laplace equation in the disk r < R satisfying the boundary condition u(R, e) = I(e) (f given) is x [ (r)n u(r, B> = 00 + ~l an Ii cos lie (20) ( r)n ] + bn R sin nO where (In' bn are the Fourier coefficients of f (see Sec. 11.I). (c) Dirichlet problem Solve the Dirichlet problem using (20) if R = I and the boundary values are u(O) = -100 volts if -7r < 0 < O. u(O) = 100 volts if 0 < e < 7r. (Sketch this disk, indicate the boundary values.) (d) Neumann problem Show that the solution of the Neumann problem y211 = 0 if r < R, llN(R, e) = f(B) (where LIN = iJ"/iJN is the directional de11vative in the direction of the outer normal) is u(r, 0) = Ao + L rn(An cos IlO + Bn sin lie) n~1 with arbitrary Ao and I TI" --n-_--cl f f(A) cos nA de, 7rIlR -TI" I " Bn = n-l f fee) sin lie de. 7rIlR _ .. (e) Compatibility condition Show that (9), Sec. 10.4, impo~es on f(O) in (d) the "compatibility condition" (f) Neumann problem Solve y 2u = 0 in the annulus I < r < 3 if liTO, 0) = sin 0, U,(3, e) = o.

The electrostatic potential satisfies Laplace's equation 'V2 11 = 0 in any region free of charges. Also the heat equation lit = C2 ,211 (Sec. 12.5) reduces to Laplace's equation if the temperature u is tinIe-independent ("steady-state case"). Using (20), find the potential (equivalently: the steady-state temperature) in the disk r < I if the boundary values are (sketch them, to see what is going on).

The electrostatic potential satisfies Laplace's equation 'V2 11 = 0 in any region free of charges. Also the heat equation lit = C2 ,211 (Sec. 12.5) reduces to Laplace's equation if the temperature u is tinIe-independent ("steady-state case"). Using (20), find the potential (equivalently: the steady-state temperature) in the disk r < I if the boundary values are (sketch them, to see what is going on).

The electrostatic potential satisfies Laplace's equation 'V2 11 = 0 in any region free of charges. Also the heat equation lit = C2 ,211 (Sec. 12.5) reduces to Laplace's equation if the temperature u is tinIe-independent ("steady-state case"). Using (20), find the potential (equivalently: the steady-state temperature) in the disk r < I if the boundary values are (sketch them, to see what is going on).

The electrostatic potential satisfies Laplace's equation 'V2 11 = 0 in any region free of charges. Also the heat equation lit = C2 ,211 (Sec. 12.5) reduces to Laplace's equation if the temperature u is tinIe-independent ("steady-state case"). Using (20), find the potential (equivalently: the steady-state temperature) in the disk r < I if the boundary values are (sketch them, to see what is going on).

The electrostatic potential satisfies Laplace's equation 'V2 11 = 0 in any region free of charges. Also the heat equation lit = C2 ,211 (Sec. 12.5) reduces to Laplace's equation if the temperature u is tinIe-independent ("steady-state case"). Using (20), find the potential (equivalently: the steady-state temperature) in the disk r < I if the boundary values are (sketch them, to see what is going on).

The electrostatic potential satisfies Laplace's equation 'V2 11 = 0 in any region free of charges. Also the heat equation lit = C2 ,211 (Sec. 12.5) reduces to Laplace's equation if the temperature u is tinIe-independent ("steady-state case"). Using (20), find the potential (equivalently: the steady-state temperature) in the disk r < I if the boundary values are (sketch them, to see what is going on).

CAS EXPERIMENT. Equipotential Lines. Guess what the equipotential lines tI(r, e) = const in Probs. 9 and 11 may look like. Then graph some of them, using partial sums of the series.

(Semidisk) Find the electrostatic potential in the semidisk r < I, 0 < e < 7r which equals I IO O( 7r - B> on the semicircle I' = I and 0 on the segment -I

Semidisk) Find the steady-state temperature in a semicircular thin plate r < a, 0 < e < 7r with the semicircle I' = a kept at constant temperature 110 and the segment -(l < X < a at O.

(Illvariance) Show that y 2 u is invariant under translations x* = x + (l, y* = Y + b and under rotations x* = x cos a - y sin a, y* = x sin a + y cos a.

(Frequency) What happens to the frequency of an eigenfunction of a dfilm if you double the tension?

(Size of a drum) A small dfilm should have a higher fundamental frequency than a large one, tension and density being the same. How does this follow from our formulas?

(Tension) Find a formula for the tension required to produce a desired fundamental frequency f I of a drum.

CAS PROJECT. Normal Modes. (a) Graph the nOimal modes 114' 115' 116 as in Fig. 306. (b) Write a program for calculating the Am's in Example 1 and extend the table to III = 15. Verify numerically that am = (Ill - ~) 7T and compute the error for 111 = 1, . . . , 10. (c) Graph the initial deflection fer) in Example 1 as well as the fIrst three partial sums of the series. Comment on accuracy. (d) Compute the radii of the nodal lines of U2' U3' 114 when R = I. How do these values compare to those of the nodes of the vibrating string of length I? Can you establish any empirical laws by experimentation with further 11m?

(Nodal lines) Is it possible that for fixed L" and R two or more II", [see (16)] with ditlerent nodal lines correspond to the same eigenvalue? (Give a reason.)

Why is Al + A2 + ... = 1 in Example I? Compute the first few partial sums until you get 3-digit accuracy. What does this problem mean in the field of music?

(Nonzero initial velocity) Show that for (17) to satisfy (9b) we must have (21) R X f rg(r)Jo(a1llrlR) dr.

(Separations) Show that substitution of II = F(r, (})G(t) into the wave equation (6), that is, gives an ODE and a PDE

Periodicity) Show that Q(8) must be periodic with period 27T and. therefore. 11 = 0, 1, 2 . in (25) and (26). Show that this yields the solutions Qn = cos 110, Qn * = sin nO, Wn = In(kr), 11 = 0, 1, ....

(Boundary condition) Show that the boundary condition (27) u(R. O. t) = 0 leads to k = kmn = amnlR, where s = amn is the mth positive zero of In(s).

(Solutions depending on both rand 8) Sho\\ that solutions of (22) satisfying (27) are (see Fig. 307) (28) CD Fig. 307. Nodal lines of some of the solutions (28)

(Initial condition) Show that IIt {r, O. 0) = 0 gives Bmn = 0, Bi;,n = 0 in (28).

Show that II~,O = 0 and UmO is identical with (16) in the current section.

(Semicircular membrane) Show that Ull represents the fundamental mode of a semicircular membrane and fInd the corresponding frequency when c2 = I and R = 1.

Derive (7) from V2 11 in Cartesian coordinates. (Show the details.)

Find the surfaces on which the functions l/I' lI2. l/3 are zero.

Sketch the functions P,,(cos c/J) for II = O. I. 2 (see (11') in Sec. 5.3).

Sketch the functions P3(COS c/J) and P 4 (cos c/J).

Verify that lin and lin * in (16) are solutions of (8).

(Dimension 3) Show that the only solution of the Laplace equation depending only on r = Vx 2 + y2 + :2 is II = elr + k with constant c and k.

(Dimension 3) Verify that 1I = dr. r = Vx 2 + y2 + :2. satisfies Laplace's equation in spherical coordinates.

(Dirichlet problem). Find the electrostatic potential between two concentric spheres of radii rl = 10 cm and r2 = 20 em kept at potentials VI = 260 V amI V2 = 110 V. respectively.

(Dimension 2. logarithmic potential) Show that the onl) solution of the two-dimensional Laplace equation depending only on r = V>;;2 + .1'2 is l/ = c In r + k with constant c and k.

(Logarithmic potential) Find the electrostatic potential between two coaxial cylinders of radii rl = 10 cm and r2 = 20 cm kept at potentials VI = 260 V and V2 = 110 V. respectively. Compare with Prob. 8. Comment.

(Heat problem) If the sUiface of the ball r2 = x 2 + y2 + :2 ~ R2 is kept at temperature zero and the initial temperature in the ball is f( 1'). show that the temperature u(r, t) in the ball is a solution of lit = c 2 (url" + 2u,./r) satisfying the conditions u(R.1) = O. u(r, 0) = fer). Show that setting v = ru gives vt = C2VJT' vCR. t) = O. vCr. 0) = rf(r). Include the condition v(O. t) = 0 (which holds because II must be bounded at r = 0). and solve the resulting problem by separating variables.

(Two-dimensional potential problems) Show that the functions x 2 - )'2. XY. xl(x2 + y2). eX cos y. r~ sin r. cos x cosh y. I~ (x2 ' + y2). and arctan <.,:Ix) satisfy Laplace's equation "xx + l/yy = O. (Two-dimensional potential problems are best solved by complex allalysis, as we shall see in Chap. 18.)

Find the potential in the interior of the sphere S: r = R = I if this interior is free of charges and the potential on Sis:

Find the potential in the interior of the sphere S: r = R = I if this interior is free of charges and the potential on Sis:

Find the potential in the interior of the sphere S: r = R = I if this interior is free of charges and the potential on Sis:

Find the potential in the interior of the sphere S: r = R = I if this interior is free of charges and the potential on Sis:

Find the potential in the interior of the sphere S: r = R = I if this interior is free of charges and the potential on Sis:

Show that in Prob. 13 the potential exterior to the sphere is the same as that of a point charge at the origin. Is this physically plausible?

Sketch the intersection of the equipotential surfaces in Prob. 14 with the xo-plane.

Find the potential exterior to the sphere in Example 2 of the text and in Prob. 15.

What is the temperature in a ball of radius I and of homogeneous material if its lower boundary hemisphere is kept at OC and its upper at 100C?

(Renection in a sphere) Let r, 0, c/J be spherical coordinates. If u(r. O. c/J) satisfies V2 11 = O. show that vCr. O. c/J) = lI( Ilr. O. c/J)lr satisfies V 2 v = O. What does this give for (l6)?

(Renection in a circle) Let r, 0 be polar coordinates. If lI(r. 0) satisfies V2 l/ = 0, show that the function v(r. 0) = lI(llr, 0) satisfies V2 v = O. What are l/ = r cos 0 and v in terms of x and y? Answer the same question for u = r2 cos 0 sin 0 and v.

TEAM PROJECT. Transmission Line and Related PDEs. Consider a long cable or telephone wire (Fig. 312) that is imperfectly insulated. so that leaks occur along the entire length of the cable. The source S of the current i(x, t) in the cable is at x = 0, the receiving end T at x = I. The current flows from S to T. through the load, and returns to the ground. Let the constants R, L. C. and G denote the resistance, inductance, capacitance to ground. and conductance to ground. respectively. of the cable per unit length (a) Show that ("first transmission line equation") au ai - - =Ri + Lax at where u(x, t) is the potential in the cable. Hint: Apply Kirchhoff's voltage law to a small portion of the cable between x and x + !n (difference of the potentials at x and x + !n = resistive drop + inductive drop). (b) Show that for the cable in (a) ("second transmission line equation"), ai au - - = Gu + Cax at Hint: Use Kirchhoff's current law (difference of the currents at x and x + LlX = loss due to leakage to ground + capacitive loss). (c) Second-order PDEs. Show that elimination of i or u from the transmission line equations leads to uxx = LCutt + (RC + GLhlt + RGu. ixx = LCitt + (RC + GL)it + RGi. (d) Telegraph equations. For a submarine cable, G is negligible and the frequencies are low. Show that this leads to the so-called submarine cable equations or telegraph equations Find the potential in a submarine cable with ends (x = 0, x = l) grounded and initial voltage distribution Va = const. (e) High-frequency line equations. Show that in the case of alternating currents of high frequencies the equations in (c) can be approximated by the so-called high-frequency line equations Solve the first of them, assuming that the initial potential is Vo sin (7fx/l). and ut(x. 0) = 0 and u = 0 at the ends x = 0 and x = I for all t.

Sketch a figure similar to Fig. 314 if c = 1 and f is "triangular" as in Example 1, Sec. 12.3.

How does the speed of the wave in Example I depend on the tension and on the mass of the string?

Verify the solution in Example 1. What traveling wave do we obtain in Example I in the case of a (nonterminating) sinusoidal motion of the left end starting at t = O?

aw aw . - + x- = x, w(x, 0) = 1, w(O, t) =

SOLVE BY LAPLACE TRANSFORMS

SOLVE BY LAPLACE TRANSFORMS w(x, 0) = 0 if x ~ 0, "'t(x, 0) = 0 if t ~ 0, w(O, t) = sin t if t ~ 0

Solve Prob. 5 by another method.

Solve Prob. 5 by another method.

Solve Prob. 5 by another method.

Solve Prob. 5 by another method.

Write down the three probably most important PDEs from memory and state their main applications.

What is the method of separating variables for PDEs? Give an example from memory.

What is the superposition principle? Give a typical application.

What role did Fourier series play in this chapter? Fourier integrals?

What are the eigenfunctions and their frequencies of the vibrating string? Of the heat equation?

What additional conditions did we consider for the wave equation? For the heat equation?

Name and explain the three kinds of boundary conditions.

What do you know about types of PDEs? About transformation to normal forms?

What do you know about types of PDEs? About transformation to normal forms?

When and why did we use polar coordinates? Spherical coordinates?

When and why did Legendre's equation occur in this chapter? Bessel's equation?

What are the eigenfunctions of the circular membrane? How do their frequencies differ in principle from those of the eigenfunctions of the vibrating string?

Explain mathematically (not physically) why we got exponential functions in separating the heat equation, but not for the wave equation.

What is the en-or function? Why did it occur and where?

Explain the idea of using Laplace transform methods for PDEs. Give an example from memory.

For what k and 111 are X4 + kx2y2 + y4 and sin nIX sinh y solutions of Laplace's equation?

Verify that (x2 - y2)/(x2 + y2)2 satisfies Laplace's equation.

Solve for II = lI(X, y):U yy + 1611 = 0

Solve for II = lI(X, y): "xx + U x - 2u = 0

Solve for II = lI(X, y): u xy + u y + x + y + ] = 0

Solve for II = lI(X, y): U yy + u y = O. u(x, 0) = J(x), lIyCX, 0) = g(x)

Find all solution u(x, y) = F(x)G(y) of Laplace's equation in two variables.

Find and sketch or graph (as in Fig. 285 in Sec. 12.3) the deflection u(x, t) of a vibrating string of length 7r, extending from x = 0 to x = 7r, and e 2 = TIp = I. starting with velocity 0 and deflection

Find and sketch or graph (as in Fig. 285 in Sec. 12.3) the deflection u(x, t) of a vibrating string of length 7r, extending from x = 0 to x = 7r, and e 2 = TIp = I. starting with velocity 0 and deflection

Find and sketch or graph (as in Fig. 285 in Sec. 12.3) the deflection u(x, t) of a vibrating string of length 7r, extending from x = 0 to x = 7r, and e 2 = TIp = I. starting with velocity 0 and deflection

Find and sketch or graph (as in Fig. 285 in Sec. 12.3) the deflection u(x, t) of a vibrating string of length 7r, extending from x = 0 to x = 7r, and e 2 = TIp = I. starting with velocity 0 and deflection

Find the temperature distribution in a laterally insulated thin copper bar (e2 = Klpu = 1.158 cm2 /sec), 50 cm long and of constant cross section with endpoints at x = 0 and 50 kept at OC and initial temperature

Find the temperature distribution in a laterally insulated thin copper bar (e2 = Klpu = 1.158 cm2 /sec), 50 cm long and of constant cross section with endpoints at x = 0 and 50 kept at OC and initial temperature

Find the temperature distribution in a laterally insulated thin copper bar (e2 = Klpu = 1.158 cm2 /sec), 50 cm long and of constant cross section with endpoints at x = 0 and 50 kept at OC and initial temperature

Find the temperature distribution in a laterally insulated thin copper bar (e2 = Klpu = 1.158 cm2 /sec), 50 cm long and of constant cross section with endpoints at x = 0 and 50 kept at OC and initial temperature

Find the temperature IItx, t) in a laterally insulated bar of length 7r. extending from x = 0 to x = 7r. with e2 = I for adiabatic boundary condition (see Problem Set 12.5) and initial temperature

Find the temperature IItx, t) in a laterally insulated bar of length 7r. extending from x = 0 to x = 7r. with e2 = I for adiabatic boundary condition (see Problem Set 12.5) and initial temperature

Find the temperature IItx, t) in a laterally insulated bar of length 7r. extending from x = 0 to x = 7r. with e2 = I for adiabatic boundary condition (see Problem Set 12.5) and initial temperature

Using partial sums, graph lI(x, t) in Prob. 33 for several constant f on conunon axes. Do these graphs agree with your physical intuition?

Let .f(x, y) = utx, y, 0) be the initial temperature in a thin square plate of side 7r with edges kept at onc and faces perfectly insulated. Separating variables. obtain from U t = C 2 V 2 U the solution x u(x, y, t) = L L Bmn sin mx sin II}" e-C2(m2+n2)t

Find the temperature in Prob. 35 if .f(x, y) = x( 7r - x)y( 7r - y).

Transform to normal form and solve (showing the details!) u xy = U xx

Transform to normal form and solve (showing the details!) lIxx + 411xy + 4uyy = 0

Transform to normal form and solve (showing the details!)U xx + 411yy = 0

Transform to normal form and solve (showing the details!) 2uxx + SUxy + 2uyy = 0

Transform to normal form and solve (showing the details!) U xx + 2l1xy + U yy = 0

Transform to normal form and solve (showing the details!) U yy + u xy - 2uxx = 0