(Variation of parameter) Another method of obtaining (4)

Solve the ODE by integration.

Solve the ODE by integration. y' = xex2/2

Solve the ODE by integration. y' cosh 4x

State the order of the ODE. Verify that the given function is a solution. (a, 17, e are arbitrary constants.) y' = 1 + y2, y = tan (x + c)

State the order of the ODE. Verify that the given function is a solution. (a, 17, e are arbitrary constants.) y" + 7T2 y = 0, Y = a cos rrx + b sin 7TX

State the order of the ODE. Verify that the given function is a solution. (a, 17, e are arbitrary constants.) y" + 2,,' + lOy = O. Y = 4e- x sin 3x

State the order of the ODE. Verify that the given function is a solution. (a, 17, e are arbitrary constants.) y' + 2y = 4(x + 1)2, Y = 5e- 2x + 2.1'2 + 2T +

State the order of the ODE. Verify that the given function is a solution. (a, 17, e are arbitrary constants.) y'" = cos x, y = -sin x + ax2 + bx + ('

Verify that y is a solution of the ODE. Determine from y the particular solution satisfying the given initial condition. Sketch or graph this solution.y' 0.5y. Y = eeO.5,,:. y(2) = 2

Verify that y is a solution of the ODE. Determine from y the particular solution satisfying the given initial condition. Sketch or graph this solution.y' = I + 4y2, Y = ~ tan (2x + e), yeO) = 0

Verify that y is a solution of the ODE. Determine from y the particular solution satisfying the given initial condition. Sketch or graph this solution. y' = y - x, y = cex + x + 1, yeO) = 3

Verify that y is a solution of the ODE. Determine from y the particular solution satisfying the given initial condition. Sketch or graph this solution. y' + 2xy = 0, y = ee-~:2. y( 1) = 1 Ie

Verify that y is a solution of the ODE. Determine from y the particular solution satisfying the given initial condition. Sketch or graph this solution. y' = y tan x, y = c sec x, yeO) = ~7T

(Existence) (A) Does the ODE y'2 = -] have a (real) solution? (B) Does the ODE 1/1 + Iyl = 0 have a general solution?

(Singular solution) An ODE may sometimes have an additional solution that cannot be obtained from the general solution and is then called a singular solution. The ODE /2 - XV' + Y = 0 is of the kind. Show by differentiation and substitution that it has the general solution y = ex - e 2 and the singular solution y = x 2/4. Explain Fig. 6.

The following problems will give you a first impression of modeling. Many more problems on modeling follow throughout this chapter. (Falling body) If we drop a stone, we can assume air resistance ("drag") to be negligible. Experiments show that under that assumption the acceleration y" = d2 Yldt2 of this motion is constant (equal to the so-called acceleration of gravity g = 9.80 mlsec2 = 32 ftlsec 2). State this as an ODE for yet), the distance fallen a~ a function of time t. Solve the ODE to get the familiar law of free fall, y = gt2/2.

(Falling body) If in Prob. 17 the stone starts at t = 0 from initial position Yo with initial velocity u = uo, show that the solution is y = gt2 /2 + uot + Yo. Hov. long does a fall of 100 m take if the body falls from rest? A fall of 200 m? (Guess first.)

(Airplane takeoff) If an airplane has a run of 3 km, statts with a speed 6 mlsec, moves wIth constant acceleration, and makes the run in I min, with what speed does it take off?

(Subsonic flight) The efficiency of the engines of subsonic airplanes depends on air pressure and usually is maximum near about 36 000 f1. Find the air pressure rex) at this height without calculation. Ph\'Sical i'!formGtioll. The ;ate of change y' (x) is propOliionai to the pressure, and at 18 000 ft the pressure has decreased to half its value )'0 at sea level.

(Half-life) The half-life of a radioactive substance is the time in which half of the given amount disappears. Hence it measures the rapidity of the decay. What is the half-life of radium 88Ra226 (in years) in Example 5?

(Interest rates) Show by algebra that the investment y(t) from a deposit Yo after t years at an imerest rate r is Ya(t) = yo[1 + r]t (Interest compounded annually) -"d(t) = .ro[l + (rl365)]365t (Interest compounded daily). Recall from calculus that [1 + (llll)r -7 e as 11 --+ x; hent:e [I + (rln)]nt --+ e't: thu~ (Interest compounded continuously). What ODE does the last function satisfy? Let the initial investment be $1000 and r = 6%. Compute the value of the investment after I year and after 5 years using each of the three formulas. [s there much difference?

Graph a direction field (by a CAS or by hand). In the field graph approximate solution curves through the given point or points (x, y) by hand. y' = eX - y. (0. 0), (0. I)

Graph a direction field (by a CAS or by hand). In the field graph approximate solution curves through the given point or points (x, y) by hand. 4yy' = -9x, (2, 2)

Graph a direction field (by a CAS or by hand). In the field graph approximate solution curves through the given point or points (x, y) by hand. y' = 1 + y2, (~1T, I)

Graph a direction field (by a CAS or by hand). In the field graph approximate solution curves through the given point or points (x, y) by hand. y' = y - 2y2. (0. 0). (0. 0.25). (0. 0.5). (0. I)

Graph a direction field (by a CAS or by hand). In the field graph approximate solution curves through the given point or points (x, y) by hand. y' = x 2 - IIy, (I, -2)

Graph a direction field (by a CAS or by hand). In the field graph approximate solution curves through the given point or points (x, y) by hand. y' = I + siny, (-1, 0), (1,4)

Graph a direction field (by a CAS or by hand). In the field graph approximate solution curves through the given point or points (x, y) by hand. V' = y3 + x 3 , (0, 1)

Graph a direction field (by a CAS or by hand). In the field graph approximate solution curves through the given point or points (x, y) by hand. y' = 2xy + I, (-I, 2), (0, 0), (1, -2)

Graph a direction field (by a CAS or by hand). In the field graph approximate solution curves through the given point or points (x, y) by hand. y' = y tanh x - 2, (-1, -2), (1,0), (1, 2)

Graph a direction field (by a CAS or by hand). In the field graph approximate solution curves through the given point or points (x, y) by hand. y' = eY/x, (I, I), (2, 2), (3, 3)

Direction fields are very useful because you can see solutions (as many as you want) without solving the ODE, which may be difficult or impossible in terms of a formula. To get a feel for the accuracy of the method, graph a field, sketch solution curves in it, and compare them with the exact solutions.

Direction fields are very useful because you can see solutions (as many as you want) without solving the ODE, which may be difficult or impossible in terms of a formula. To get a feel for the accuracy of the method, graph a field, sketch solution curves in it, and compare them with the exact solutions.

Direction fields are very useful because you can see solutions (as many as you want) without solving the ODE, which may be difficult or impossible in terms of a formula. To get a feel for the accuracy of the method, graph a field, sketch solution curves in it, and compare them with the exact solutions.

Direction fields are very useful because you can see solutions (as many as you want) without solving the ODE, which may be difficult or impossible in terms of a formula. To get a feel for the accuracy of the method, graph a field, sketch solution curves in it, and compare them with the exact solutions.

Direction fields are very useful because you can see solutions (as many as you want) without solving the ODE, which may be difficult or impossible in terms of a formula. To get a feel for the accuracy of the method, graph a field, sketch solution curves in it, and compare them with the exact solutions.

A body moves on a straight line, with velocity as given. and yet) is its distance from a fixed point 0 and t time. Find a model of the motion (an ODE). Graph a direction field. 12 CHAP. 1 First-Order ODEs In it sketch a solution curve corresponding to the given initial condition.

A body moves on a straight line, with velocity as given. and yet) is its distance from a fixed point 0 and t time. Find a model of the motion (an ODE). Graph a direction field. 12 CHAP. 1 First-Order ODEs In it sketch a solution curve corresponding to the given initial condition. Velocity equal to the reciprocal ofthe distance, y(l) = I Product of velocity and distance equal to -t, y(3) = -3

A body moves on a straight line, with velocity as given. and yet) is its distance from a fixed point 0 and t time. Find a model of the motion (an ODE). Graph a direction field. 12 CHAP. 1 First-Order ODEs In it sketch a solution curve corresponding to the given initial condition. Velocity plus distance equal to the square of time, yeO) = 6

(Skydiver) Two forces act on a parachutist, the attraction by the earth mg (/1/ = mass of person plus equipment. g = 9.8 m/sec2 the acceleration of gravity) and the air resistance, assumed to be proportional to the square of the velocity vet). Using Newton's second law of motion (mass X acceleration = resultant of the forces), set up a model (an ODE for v(t. Graph a direction field (choosing III and the constant of proportionality equal to 1). Assume that the parachute opens when v = 10m/sec. Graph the corresponding solution in the field. What is the limiting velocity?

CAS PROJECT. Direction Fields. Discuss direction fields as follows. (a) Graph a direction field for the ODE y' = I - Y and in it the solution satisfying yeO) = 5 showing exponential approach. Can you see the limit of any solution directly from the ODE? For what initial condition will the solution be increasing? Constant? Decreasing? (b) What do the solution curves of y' = _X3/y3 look like, as concluded from a direction field. How do they seem to differ from circles? What are the isoclines? What happens to those curves when you drop the minus on the right? Do they look similar to familiar curves? First. guess. (c) Compare. as best as you can, the old and the computer methods, their advantages and disadvantages. Write a short report.

(Constant of integl'3tion) An arbitrary constant of integration must be introduced inunediately when the integration is performed. Why is this important? Give an example of your own.

Find a general solution. Show the steps of derivation. Check your answer by substitution.y' + (x + 2)"2 = 0

Find a general solution. Show the steps of derivation. Check your answer by substitution. y' = 2 sec 2y

Find a general solution. Show the steps of derivation. Check your answer by substitution. y' = {y + 9X)2 (y + 9\" = v)

Find a general solution. Show the steps of derivation. Check your answer by substitution.)'y' + 36x = 0

Find a general solution. Show the steps of derivation. Check your answer by substitution.y' = (4x2 + y2)/(xy)

Find a general solution. Show the steps of derivation. Check your answer by substitution.y' sin TTX = Y cos 7TX

Find a general solution. Show the steps of derivation. Check your answer by substitution.xy' = i)"2 + Y

Find a general solution. Show the steps of derivation. Check your answer by substitution. y' e""x = )'2 + I

Find the particular solution. Show the steps of derivation, beginning with the general solution. (L, R, b are constants.)yy' + 4x = 0, y(O) = 3

Find the particular solution. Show the steps of derivation, beginning with the general solution. (L, R, b are constants.) dr/dt = -2tr, r(O) = ro

Find the particular solution. Show the steps of derivation, beginning with the general solution. (L, R, b are constants.)2xyy' = 3)"2 + x 2 , )"(1) = 2

Find the particular solution. Show the steps of derivation, beginning with the general solution. (L, R, b are constants.) L d/ldt + RI = 0, 1(0) = 10

Find the particular solution. Show the steps of derivation, beginning with the general solution. (L, R, b are constants.)v' = vlx + (2x 3 /v) COS(X2), y(v.;;:t2) = v;-

Find the particular solution. Show the steps of derivation, beginning with the general solution. (L, R, b are constants.) >xy"= 2(x + 2iy 3, yeO) = l/Vs = 0.45

Find the particular solution. Show the steps of derivation, beginning with the general solution. (L, R, b are constants.) x.v' = y + 4x 5 cos2(y/x), H2) = 0

Find the particular solution. Show the steps of derivation, beginning with the general solution. (L, R, b are constants.)y'x Inx = y, ."(3) = In 81

Find the particular solution. Show the steps of derivation, beginning with the general solution. (L, R, b are constants.)dr/dO = b[(dr/dO) cos 0 + r sin 0], r(i7T) 7T. 0< b < I

Find the particular solution. Show the steps of derivation, beginning with the general solution. (L, R, b are constants.) yy' = (x - l)e- y2, y(O) = 1

(Particulal' solution) Introduce limits of integration in (3) such that y obtained from (3) satisfies the initial condition y(xo) = )'0' Try the formula out on Prob. 19.

(Curves) Find all curves in the xy-plane whose tangents all pass through a given point (a, b).

(Cm'ves) Show that any (nonverticaD straight line through the origin of the xy-plane intersects all solution curves of y' = g()'/x) at the same angle.

(Exponential growth) If the growth rate of the amount of yeast at any time t is proportional to the amount present at that time and doubles in I week, how much yeast can be expected after 2 weeks? After 4 weeks?

(Population model) If in a population of bacteria the birth rate and death rate are proportional to the number of individuals present, what is the popUlation as a function of time? Figure out the limiting situation for increasing time and interpret it.

(Radiocal'bon dating) If a fossilized tree is claimed to be 4000 years old. what should be its 6C14 content expressed as a percent of the ratio of 6C14 to 6C12 in a living organism?

(Gompel'tz gl'Owth in tumors) TIle Gompertz model is r' = -Av In \' (A > 0), where yet) is the mass of tu~or cells' at time t. The model agrees well with clinical observations. The declining growth rate with increasing y > 1 corresponds to the fact that cells in the interior of a tumor may die because of insufficient oxygen and nutrients. Use the ODE to discuss the growth and decline of solutions (tumors) and to find constant solutions. Then solve the ODE.

(Dl-yel') If wet laundry loses half of its moisture during the first 5 minutes of drying in a dryer and if the rate of loss of moisture is proportional to the moisture content, when will the laundry be practically dry, say, when will it have lost 95% of its moisture? First guess.

(Alibi?) Jack, arrested when leaving a bar, claims that he has been inside for at least half an hour (which would provide him with an alibi). The police check the water temperature of his car (parked near the entrance of the bar) at the instant of arrest and again 30 minutes later, obtaining the values 190F and 110F, respectively. Do these results give Jack an alibi? (Solve by inspection.)

(Law of cooling) A thermometer, reading 10C, is brought into a room whose temperature is 23C, Two minutes later the thermometer reading is 18C, How long will it take until the reading is practically 23C, say, 22.8C? First guess.

(TonicelIi's law) How does the answer in Example 5 (the time when the tank is empty) change if the diameter of the hole is doubled? First guess.

(TolTicelli's law) Show that (7) looks reasonable inasmuch as V2gh(t) is the speed a body gains if it falls a distance h (and air resistance is neglected).

(Rope) To tie a boat in a harbor. how many times must a rope be wound around a bollard (a vertical rough cylindrical post fixed on the ground) so that a man holding one end of the rope can resist a force exerted by the boat one thousand times greater than the man can exert? First guess. Experiments show that the change /1S of the force S in a small portion of the rope is proportional to S and to the small angle /1 in Fig. 13, Take the proportionality constant 0.15,

(Mixing) A tank contains 800 gal of water in which 200 Ib of salt is dissolved. Two gallons of fresh water rLms in per minute, and 2 gal of the mixture in the tank. kept uniform by stirring, runs out per minute. How much salt is left in the tank after 5 hours?

WRITING PROJECT. Exponential Increase, Decay, Approach. Collect. order, and present all the information on the ODE y' = ky and its applications fi'om the text and the problems. Add examples of your own.

CAS EXPERIMENT. Graphing Solutions. A CAS can usually graph solutions even if they are given by integrals that cannot be evaluated by the usual methods of calculus. Show this as follows.(A) Graph the curves for the seven initial value problems y' = e-x2/ 2 , yeO) = 0, I, 2, 3, common axes. Are these curves congment? Why? (B) Experiment with approximate curves of nth partial sums of the Maclaurin series obtained by term wise integration of that of y in (A); graph them and describe qualitatively the accuracy for a fixed interval o ~ x ~ b and increasing n, and then for fixed nand increasing b. (C) Experiment with y' = cos (x2 ) as in (8). (D) Find an initial value problem with solution y = e-2 L"e-t2 dt and experiment with it as in (8).

TEAM PROJECT. Tonicelli's Law. Suppose that the tank in Example 5 is hemispherical, of radius R, initially full of water, and has an outlet of 5 cm2 crosssectional area at the bottom. (Make a sketch.) Set up the model for outflow. Indicate what portion of your work in Example 5 you can use (so that it can become part of the general method independent of the shape of the tank). Find the time t to empty the tank (a) for any R, (b) for R = I m. Plot t as function of R. Find the time when h = R/2 (a) for any R, (b) for R = 1 m.

Test for exactness. If exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is given, determine the corresponding particular solu[ion. x 3 dx + )'3 dy = 0

Test for exactness. If exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is given, determine the corresponding particular solu[ion. (x - y)(dx - dy) = 0

Test for exactness. If exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is given, determine the corresponding particular solu[ion. 71' sin 71'X sinh)' dx + cos 71'X cosh y dy = 0

Test for exactness. If exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is given, determine the corresponding particular solu[ion. (e Y - ye X ) dx + (xe Y - eX) dy = 0

Test for exactness. If exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is given, determine the corresponding particular solu[ion. 9x dx + 4y dy = 0

Test for exactness. If exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is given, determine the corresponding particular solu[ion. eX(cos y dx - sin y dy) = 0

Test for exactness. If exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is given, determine the corresponding particular solu[ion. e-28 dr - 2re-28 d() = 0

Test for exactness. If exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is given, determine the corresponding particular solu[ion. (2x + 11)' - ylx2) dx + (2y + 1Ix - xly2) dy = 0

Test for exactness. If exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is given, determine the corresponding particular solu[ion. (-ylx 2 + 2 cos 2x) dx + (lIx - 2 sin 2y) dy = 0

Test for exactness. If exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is given, determine the corresponding particular solu[ion. - 2xy sin (X2 ) dx + cos (x 2 ) dy = 0

Test for exactness. If exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is given, determine the corresponding particular solu[ion. - y dx + x dy = 0

Test for exactness. If exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is given, determine the corresponding particular solu[ion. (e x + y - y) dx + (xex + y + 1) dy = 0

Test for exactness. If exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is given, determine the corresponding particular solu[ion. -3\" dx + 2x dy = O. F(x. y) = ylx4

Test for exactness. If exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is given, determine the corresponding particular solu[ion. (x 4 + )'2) dx - xv dy = 0, ),(2) = I

Test for exactness. If exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is given, determine the corresponding particular solu[ion. e 2X(2 cos y dx - sin y dy) = 0, y(O) 0

Test for exactness. If exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is given, determine the corresponding particular solu[ion. -sinxy (y dx + x dy) = 0, y(l) = 'iT

Test for exactness. If exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is given, determine the corresponding particular solu[ion. (cos wX + w sin wx) dx + eX dy = O. y(Q) = 1

Test for exactness. If exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is given, determine the corresponding particular solu[ion. (cos xy + xly) dx + (1 + (xly) cos xy) dy = 0

Test for exactness. If exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is given, determine the corresponding particular solu[ion. e-Y dx + e-X(-e-Y + 1) dy = 0, F = eX+ Y

Test for exactness. If exact, solve. If not, use an integrating factor as given or find it by inspection or from the theorems in the text. Also, if an initial condition is given, determine the corresponding particular solu[ion. (sin y cos y + X cos2 y) dx + x dy = 0

Under what conditions for the constants A, B. C, D is (Ax + By) dx + (ex + Dy) dy = 0 exact? Solve the exact equation.

Graph paI1icuiar solutions of the following ODE. proceeding as explained. (d) [n another graph show the solution curves satisfying y(O) = ::':::1. ::':::2, ::':::3. ::':::4. Compare the quality of (c) and (d) and comment. (21) I Y cos x dx + - dy = 0 (a) Test for exactness. If nece~sary, find an integrating factor. Find the general solution II(X. y) = c. Solve (21) by separating variables. Is this simpler than (a)? (c) Graph contours II(X, y) = c by your CAS. (Cf. Fig. 16.) (d) [n another graph show the solution curves satisfying y(O) = ::':::1. ::':::2, ::':::3. ::':::4. Compare the quality of (c) and (d) and comment. (e) Do the same steps for another nonexact ODE of your choice.

WRITING PROJECT. Working Backward. Start from solutions u(x, v) = c of your choice. find a corresponding exact ODE, destroy exactness by a multiplication or division. This should give you a feel for the form of ODEs you can reach by the method of integrating factors. (Working backward is useful in other areas, too: Euler and other great masters frequently did it.l

TEAM PROJECT. Solution by Several Methods. Show this as indicated. Compare the amount of work. (A) eY(sinh x dx + cosh x dy) = 0 as an exact ODE and by separation. (B) (I + 2x) cos y dx + dy!cos y = 0 by Theorem 2 and by separation. (C) (x2 + y2) eLI: - 2xy dy = 0 by Theorem I or 2 and by separation with v = ylx. (D) 3x2 .v dx + 4x3 dy = 0 by Theorems I and 2 and by separation. (E) Search the text and the problems for further ODEs that can be solved by more than one of the methods discussed so far. Make a list of these ODEs. Find further cases of your own.

(CAUTION!) Show that e-1n x = l/x (not -x) and e-1n(sec x) = cos x.

(Integration constant) Give a reason why in (4) you may choose the constant of integration in fp dx to be zero.

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.) y' + 3.5y = 2.8

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.) y' = 4y + x

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.) )" + 1.25y = 5, yeO) 6.6

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.) x 2 y' -t- 3xy = lIx, y(l) = -1

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.) y' + ky = e 2kx

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.) y' + 2y = 4 cos 2x, y(!7T) = 2

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.) y' 6(y - 2.5) tanh 1.5x

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.) y' + 4x 2 y = (4x 2 - x)e- x2/ 2

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.) )" + 2y sin 2x = 2ecos 2X, yeO) 0

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.) )" tan x = 2y - 8, Y(~7T) = 0

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.) y' + 4y cot 2x = 6 cos 2x, y(!7T) = 2

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.) y' + )' tan x = e- O. Ob cos x, 1'(0) = 0

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.) y' + Y/X2 = 2xe 1/ x , y(l) = 13.86

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.) y' cos 2 x + 3y = I, y(!7T) = ~

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.) x 3 y' + 3x2 y = 5 sinh lOx

Using a method of this section or separating variables, find the general solution. If an initial condition is given, find also the particular solution and sketch or graph it. y' + Y = y2, yeO) = -I

Using a method of this section or separating variables, find the general solution. If an initial condition is given, find also the particular solution and sketch or graph it. y' = 5.7y - 6.5 y 2

Using a method of this section or separating variables, find the general solution. If an initial condition is given, find also the particular solution and sketch or graph it. (x2 + I)y' = -tan y. y(O) = ~7T

Using a method of this section or separating variables, find the general solution. If an initial condition is given, find also the particular solution and sketch or graph it.y' + (x + I)y = eX 'y3, y(O) = 0.5

Using a method of this section or separating variables, find the general solution. If an initial condition is given, find also the particular solution and sketch or graph it. y' sin 2y + x cos 2y = 2x

Using a method of this section or separating variables, find the general solution. If an initial condition is given, find also the particular solution and sketch or graph it. 2yy' + .\'2 sin x = sin x. yeO) = V2

Using a method of this section or separating variables, find the general solution. If an initial condition is given, find also the particular solution and sketch or graph it. y' + x 2y = (e- X3 sinh X)/(3y2)

(Investment programs) Bill opens a retirement savings account with an initial amount Yo and then adds $k to the account at the beginning of every year until retirement at age 65. Assume that the interest is compounded continuously at the same rate R over the years. Set up a model for the balance in the account and find the general solution as well as the particular solution, letting t = 0 be the instant when the account is opened. How much money will Bill have in (he account at age 65 if he starts at 25 and invests $1000 initially as well as annually, and the interest rate R is 6%? How much should he invest initially and annually (same amounts) to obtain the same final balance as before if he starts at age 45? First, guess.

(Mixing problem) A tank (as in Fig. 9 in Sec. 1.3) contains 1000 gal of water in which 200 Ib of salt is dissolved. 50 gal of brine. each uallon containin u (I + cos t) Ib of dissolved salt, run: into the tank pe~ minute. The mixture. kept unifonn by stirring. nms out at the same rate. Find the amount of salt in the tank at any time t (Fig. 20).

(Lake Erie) Lake Erie has a water volume of about 450 km3 and a flow rate (in and out) of about 175 km3 per year. If at some instant the lake has pollution c~nc~ntration p = 0.04%, how long, approximately. wIll It take to decrease it to pl2. assuming that the inflow is much cleaner, say, it has pollution concentration p/4. and the mixture is unifonn (an assumption that is only very imperfectly true)? First, guess.

(Heating and cooling of a building) Heating and cooling of a building can be modeled by the ODE T' = k 1(T - Ta) + k2(T - Tw) + P, where T = T(t) is the temperature in the building at time t, Ta the outside temperature, T w the temperature wanted in the building. and P the rate of increase of T due to machines and people in the building, and kl and ~ are (negative) constants. Solve this ODE, assuming P = canst, T w = canst, and To varying sinusoidally over 24 hours, say, Ta = A - C cos (27T/2A)t. Discuss the effect of each tenn of the equation on the solution.

(Drug injection) Find and solve the model for drug injection into the bloodstream if, beginning at t = 0, a constant amount A g/min is injected and the drug is simultaneously removed at a rate proportional to the amount of the drug present at time t.

(Epidemics) A model for the spread of contagious diseases is obtained by assuming that the rate of spread is proportional to the number of contacts between infected and noninfected persons, who are assumed to move freely among each other. Set up the model. Find the equilibrium solutions and indicate their stability or instability. Solve the ODE. Find the limit of the proportion of infected persons as t -+ x and explain what it means.

(Extinction vs. unlimited growth) If in a population y(t) the death rate is proportional to the population, and the birth rate is proportional to the chance encounters of meeting mates for reproduction. what will the model be? Without solving. find out what will eventually happen to a small initial population. To a large one. Then solve the model.

(Harvesting renewable resources. Fishing) Suppose that the population yU) of a certain kind of fish is given by the logistic equation (8), and fish are caught at a rate Hy proportional to y. Solve this so-called Schaefer model. Find the equilibrium solutions Yl and Y2 (> 0) when H < A. The expression Y = HY2 is called the equilibrium harvest or sustainable yield corresponding to H. Why?

(Harvesting) In Prob. 32 find and graph the solution satisfying yeO) = 2 when (for simplicity) A = B = I and H = 0.2. What is the limit? What does it mean? What if there were no fishing?

(Intermittent harvesting) In Prob. 32 assume that you fish for 3 years, then fishing is banned for the next 3 years. Thereafter you start again. And so on. This is called illtermittent /Uln'estillg. Describe qualitatively how the population will develop if intermitting is continued periodically. Find and graph the solution for the first 9 years, assuming that A = B = I, H = 0.2, and yeO) = 2.

(Harvesting) If a population of mice (in multiples of 1000) follows the logistic law with A = I and B = 0.25. and if owls catch at a time rate of 10% of the population present, what is the model, its equilibrium harvest for that catch. and its solution?

(Harvesting) Do you save work in Prob. 34 if you first transform the ODE to a linear ODE? Do this transformation. Solve the resulting ODE. Does the resulting yet) agree with that in Prob. 34?

The sum YI + Y2 of two solutions YI and Y2 of the homogeneous equation (2) is a solution of (2), and so is a scalar mUltiple aYI for any constant a. These properties are not true for (1)1

Y = 0 (that is, .v(x) = 0 for all x, also written y(x) "'" 0) is a solution of (2) [not of (I) if rex) =1= 01], called the trivial solution.

The sum of a solution of (I) and a solution of (2) is a solution of (1).

The difference of two solutions of (l) is a solution of (2).

If Yl is a sulution of (I), what can you say about eYl?

If YI and Y2 are solutions of y~ + PYI = rl and Y~ + PY2 = r2, respectively (with the same p!), what can you say about the sum.vl + Y2?

CAS EXPERIMENT. (a) Solve the ODE y' - ylx = -x-1 cos (l/x). Find an initial condition for which the arbitrary constanl is zero. Graph the resulting particular solution, experimenting to obtain a good figure near x = O. (b) Generalizing (a) from 11 = I to arbitrary 11, solve the ODE y' - nylx = _xn - 2 cos (llx). Find an initial condition as in (a). and experiment with the graph.

TEAM PROJECT. Riccati Equation, Clairaut Equation. A Riccati equation is of the form (1 I) y' + p(x)y = gp:)y 2 + hex). A Clairaut equation is of the form (12) y = xy' + g(y'). (a) Apply the transformation y = Y + lilt to the Riccati equation (1 I ), where Y is a solution of (11), and oblain for u the linear ODE u' + (2Yg - P)U = -g. Explain the effect of the transformation by writing it as y = Y + v, v = lilt. (b) Show that Y = Y = x is a solution of y' - (2x 3 + l)y = _x2y2 - X4 - X + and solve this Riccati equation. showing the details. (c) Solve y' + (3 - 2X2 sin x)y = _y2 sin x + 2x + 3x2 - X4 sin x, using (and Verifying) that y = x 2 is a solution. (d) By working "backward" from the L1-equation find further Riccati equations that have relatively simple solutions. (e) Solve the Clairautequationy = xy' + 1/y'.Hillt. Differentiate this ODE with respect to x. (f) Solve the Clairaut equation /2 - xy' + Y = 0 in Prob. 16 of Problem Set 1.1. (g) Show that the Clairaut equation (12) has as solutions a family of straight lines Y = ex + gee) and a singular solution determined by g' (s) = -x, where s = y', that forms the envelope of that family.

(Variation of parameter) Another method of obtaining (4) results from the following idea. Write (3) as ey*, where y* is the exponential function. which is a solution of the homogeneous linear ODE y*' + py* = O. Replace the arbitrary constant e in (3) with a function II to be determined so that the resulting function y = IIY* is a solution of the nonhomogeneous linear ODE y' + PY = r.

TEAM PROJECT. Transformations of ODEs. We have transformed ODEs to separable form, to exact form, and to linear form. The purpose of such transfonnations is an extension of solution methods to larger classes of ODEs. Describe the key idea of each of these transformations and give three typical examples of your choice for each transformation, shOwing each step (not just the transformed ODE)

Sketch or graph some of the given curves. Guess what their orthogonal trajectories may look like. Find these trajectories. Y = 4x + c

Sketch or graph some of the given curves. Guess what their orthogonal trajectories may look like. Find these trajectories. y = clx

Sketch or graph some of the given curves. Guess what their orthogonal trajectories may look like. Find these trajectories. y = ex

Sketch or graph some of the given curves. Guess what their orthogonal trajectories may look like. Find these trajectories. y2 = 2X2 + c

Sketch or graph some of the given curves. Guess what their orthogonal trajectories may look like. Find these trajectories. x 2 y = C

Sketch or graph some of the given curves. Guess what their orthogonal trajectories may look like. Find these trajectories. y = ce- 3x

Sketch or graph some of the given curves. Guess what their orthogonal trajectories may look like. Find these trajectories. Y = cex2/2

Sketch or graph some of the given curves. Guess what their orthogonal trajectories may look like. Find these trajectories. x 2 - y2 = c

Sketch or graph some of the given curves. Guess what their orthogonal trajectories may look like. Find these trajectories. 4x 2 + y2 = e

Sketch or graph some of the given curves. Guess what their orthogonal trajectories may look like. Find these trajectories. x = cVy

Sketch or graph some of the given curves. Guess what their orthogonal trajectories may look like. Find these trajectories. x = ceyl4

Sketch or graph some of the given curves. Guess what their orthogonal trajectories may look like. Find these trajectories. x 2 + (y - c)2 = c 2

(y as independent variable) Show that (3) may be written dx/dy = -f(x, y). Use this form to find the orthogonal trajectories of y = 2x + ce-x.

(Family g(x,y) = c) Show that if a family is given as g(x, y) = c, then the orthogonal trajectories can be obtained from the following ODE, and use the latter to solve Prob. 6 written in the form g(x, y) = c. dy ag/ay dx ag/iJx

(Cauchy-Riemann equations) Show that for a family u(x, y) = c = const the orthogonal trajectories vex, y) = c* = const can be obtained from the following Cauchy-Riemann equations (which are basic in complex analysis in Chap. 13) and use them to find the orthogonal trajectories of eX sin y = const. (Here, subscripts denote partial derivatives.)

(Fluid flow) Suppose that the streamlines of the flow lpaths of the particles of the fluid) in Fig. 24 are 'ltlx, y) = xy = COllst. Find their orthogonal trajectories (called equipotential lines, for reasons given in Sec. 18.4).

(Electric field) Let the electric equipotential lines (curves of constant potential) between two concentric cylinders (Fig. 22) be given by u(x, y) = x2 + y2 = c. Use the method in the text to find their orthogonal trajectories (the curves of electric force).

(Electric field) The lines of electric force of two opposite charges of the same strength at (-1. 0) and (1, 0) are the circles through (-1. 0) and (l, 0). Show that these circles are given by x 2 + (y - c)2 = 1 -+ c2. Show that the equipotential lines (orthogonal trajectories of those circles) are the circles given by lx + C*)2 + )'2 = C*2 - I (dashed in Fig. 25).

(Temperature field) Let the isotherms (curves of constant temperature) in a body in the upper half-plane y > 0 be given by 4X2 + 9)"2 = c. Find the orthogonal trajectories lthe curves along which heat will flow in regions filled with heat-conducting material and free of heat sources or heat sinks).

TEAM PROJECT. Conic Sections. (A) State the main steps of the present method of obtaining orthogonal trajectorics. (B) Find conditions under which the orthogonal trajectories of families of ellipses x 2 /a2 + y2/b2 = C are again conic sections. Illustrate your result graphically by sketches or by using your CAS. What happens if a~ O?If b~ O? (C) Investigate families of hyperbolas x 2 /a2 - y2/b2 = c in a similar fashion. (D) Can you find more complicated curves for which you get ODEs that you can solve? Give it a try.

Vertical strip) If the a~sumptions of Theorems I and 2 are satisfied not merely in a rectangle but in a vertical infinite strip Ix - xol < a, in what interval will the solution of (1) exist?

(Existence?) Does the initial value problem (x - l)y' = 2y, y(l) = I have a solution? Does your result contradict our present theorems?

(Common points) Can two solution curves of the same ODE have a common point in a rectangle in which the assumptions of the present theorems are satisfied?

(Change of initial condition) What happens in Prob. 2 if you replace yO) = 1 with yO) = k?

(Linear ODE) If p and I' in y' + p(x)y = rex) are continuous for all x in an interval Ix - xol ~ a, show that f(x, y) in this ODE satisfies the conditions of our present theorems, so that a cOlTesponding initial value problem has a unique solution. Do you actually need these theorems for this ODE?

(Three possible cases) Find all initial conditions such that lx2 - 4x)y' = (2x - 4)y has no solution, precisely one solution, and more than one solution.

(Length of x-interval) In most cases the solution of an initial value problem (1) exists in an x-interval larger than that guaranteed by the present theorems. Show this fact for y' = 2y2, yO) = I by finding the best possible a (choosing b optimally) and comparing the result with the actual solution.

PROJECT. Lipschitz Condition. (A) State the definItion of a Lipschitz condition. Explain its relation to the existence of a partial derivative. Explain its significance in our present context. Illustrate your statements by examples of your own. B) Show that for a lillear ODE y' + p(:.:)y = rex) with continuous p and r in Ix - xol ~ a a Lipschitz condition holds. This is remarkable because it means that for a linear ODE the continuity of f(x, y) guarantees not only the existence but also the uniqueness of the solution of an initial value problem. (Of course, this also follows directly from (4) in Sec. 1.5.) (C) Discuss the uniqueness of solution for a few simple ODEs that you can solve by one of the methods considered, and find whether a Lipschitz condition is satisfied.

Maximum a) What IS the largest possible a In Example I in the text?

CAS PROJECT. Picard Iteration. (A) Show that by integrating the ODE in (I) and observing the initial condition you obtain (6) y(x) = Yo + f f(t. \'(t dt.

Explain the tenns ordinary d~fferellfial equatiol/ (ODE). partial d~fferellfial equation (PDE). order. gel/eral solution. and particular solutioll. Give examples. Why are these concepts of imp0l1ance?

What is an initial condition? How is this condition used in an initial value problem?

What is a homogeneous linear ODE? A nonhomogeneous linear ODE? Why are these equations simpler than nonlinear ODEs?

What do you know about direction fields and their practical importance?

Give examples of mechanical problems that lead to ODEs.

Why do electric circuits lead to ODEs?

Make a list of the solution methods considered. Explain each method with a few short sentences and illustrate it by a typical example.

Can certain ODEs be solved by more than one method? Give three examples.

What are integrating factors? Explain the idea. Give examples.

Does every first-order ODE have a solution? A general solution? What do you know about uniqueness of solutions?

Graph a direction field (by a CAS or by hand) and sketch some of the solution curves. Solve the ODE exactly and compare. ,.' = 1 + 4)'2

Graph a direction field (by a CAS or by hand) and sketch some of the solution curves. Solve the ODE exactly and compare. y' 3y - 2~

Graph a direction field (by a CAS or by hand) and sketch some of the solution curves. Solve the ODE exactly and compare.

Graph a direction field (by a CAS or by hand) and sketch some of the solution curves. Solve the ODE exactly and compare.

Find the general solution. Indicate which method in this chapter you are using. Show the details of your work. y' = x 2 (1 + .1'2)

Find the general solution. Indicate which method in this chapter you are using. Show the details of your work. y' = x(y - x 2 + I)

Find the general solution. Indicate which method in this chapter you are using. Show the details of your work. yy' + xy2 = x

Find the general solution. Indicate which method in this chapter you are using. Show the details of your work. -7T sin TTX cosh 3y dx + 3 cos TTX sinh 3y dy = 0

Find the general solution. Indicate which method in this chapter you are using. Show the details of your work. y' + Y sin x = sin x

Find the general solution. Indicate which method in this chapter you are using. Show the details of your work. y' - y = 1Iy

Find the general solution. Indicate which method in this chapter you are using. Show the details of your work. sin 2y dx + 2x cos 2y dy = 0

Find the general solution. Indicate which method in this chapter you are using. Show the details of your work.x/ = x tan (ylx) + Y

Find the general solution. Indicate which method in this chapter you are using. Show the details of your work.y cos xy - 2x) dx + lx cos x)' + 2y) dy = 0

Find the general solution. Indicate which method in this chapter you are using. Show the details of your work.xy' = (y - 2X)2 +)' (Set.\ - 2x = z.)

Find the general solution. Indicate which method in this chapter you are using. Show the details of your work. sin (y - x) dx + [cos (y - x) - sin (y - x)] dy = 0

Find the general solution. Indicate which method in this chapter you are using. Show the details of your work. xy' = (ylx)3 + y

Solve the following initial value problems. Indicate the method used. Show the details of your work. yy' + x = O. y(3) = 4

Solve the following initial value problems. Indicate the method used. Show the details of your work. y' 3y=-12y 2. y(O)=2

Solve the following initial value problems. Indicate the method used. Show the details of your work. .v' = I + )'2, Y(~7T) = 0

Solve the following initial value problems. Indicate the method used. Show the details of your work. y' + 7T)' = 2b cos 7TX, yeO) = 0

Solve the following initial value problems. Indicate the method used. Show the details of your work. (2xy2 - sin x) dx + (2 + 2X2)') dy = o. yeO) = I

Solve the following initial value problems. Indicate the method used. Show the details of your work. [2y + y 2 1x + eX(l + llx)] dx + lx + 2y) dy = 0, y(l) = I

~Heat flow) If the isothelms in a region are x 2 - )'2 = c, what are the curves of heat flow (assuming orthogonality)?

(Law of cooling) A thennometer showing WaC is brought into a room whose temperature is 25C. After 5 minutes it shows 20C. When will the thennometer practically reach the room temperature, say, 24.9C?

(Half-life) If 10o/c of a radioactive substance disintegrates in 4 days, what is it~ half-life?

(HaIf-life) What is the half-life of a substance if after 5 days, 0.020 g is present and after 10 days, 0.015 g?

(HaIf-life) When will 99% of the substance in Prob. 35 have disintegrated?

(Air circulation) In a room containing 20000 ft3 of air, 600 ft3 of fresh air tlows in per minute, and the mixture (made practically unifonn by circulating fans) is exhausted at a rate of 600 cubic feet per minute (cfm). What is the amount of fresh air Yl1) at any time if yeO) = 0'1 After what time will 90% of the air be fresh?

(Electric field) If the equipotential lines in a region of the x)"-plane are 4x2 + y2 = c, what are the curves of the electIical force? Sketch both families of curves.

(Chemistry) In a bimolecular reaction A + B -? M, a moles per liter of a substance A and b moles per liter of a substance B are combined. Under constant temperature the rate of reaction is y' = k(a - y)(b - .1') (Law of mass action); that is, y' is proportional to the product of the concentrations of the substances that are reacting. where )'(t) is the number of moles per liter which have reacted after time t. Solve this ODE, assuming that a *- b.

(Population) Find the population y(1) if the birth rate is proportional to y(r) and the death rate is proportional to the square of y( f).

(Curves) Find all curve~ in the first quadrant of the Ayplane such that for every tangent. the segment between the coordinate axes is bisected by the point of tangency. (Make a sketch.)

(Optics) Lambert's law of absorption9 states that the absorption of light in a thin transparent layer is proportional to the thickness of the layer and to the amount of light incident on that layer. Formulate this law as an ODE and solve it.