 CHAPTER 1.1.1: Solve the ODE by integration.
 CHAPTER 1.1.2: Graph a direction field (by a CAS or by hand). In the field graph a...
 CHAPTER 1.1.3: (Constant of integl'3tion) An arbitrary constant of integration mus...
 CHAPTER 1.1.4: Test for exactness. If exact, solve. If not, use an integrating fac...
 CHAPTER 1.1.5: (CAUTION!) Show that e1n x = l/x (not x) and e1n(sec x) = cos x.
 CHAPTER 1.1.6: Sketch or graph some of the given curves. Guess what their orthogon...
 CHAPTER 1.1.7: Vertical strip) If the a~sumptions of Theorems I and 2 are satisfie...
 CHAPTER 1.1: Explain the tenns ordinary d~fferellfial equatiol/ (ODE). partial d...
 CHAPTER 1.1.2: Graph a direction field (by a CAS or by hand). In the field graph a...
 CHAPTER 1.1.3: Find a general solution. Show the steps of derivation. Check your a...
 CHAPTER 1.1.4: Test for exactness. If exact, solve. If not, use an integrating fac...
 CHAPTER 1.1.5: (Integration constant) Give a reason why in (4) you may choose the ...
 CHAPTER 1.1.6: Sketch or graph some of the given curves. Guess what their orthogon...
 CHAPTER 1.1.7: (Existence?) Does the initial value problem (x  l)y' = 2y, y(l) = ...
 CHAPTER 1.2: What is an initial condition? How is this condition used in an init...
 CHAPTER 1.1.1: Solve the ODE by integration. y' = xex2/2
 CHAPTER 1.1.2: Graph a direction field (by a CAS or by hand). In the field graph a...
 CHAPTER 1.1.3: Find a general solution. Show the steps of derivation. Check your a...
 CHAPTER 1.1.4: Test for exactness. If exact, solve. If not, use an integrating fac...
 CHAPTER 1.1.5: Find the general solution. If an initial condition is given, find a...
 CHAPTER 1.1.6: Sketch or graph some of the given curves. Guess what their orthogon...
 CHAPTER 1.1.7: (Common points) Can two solution curves of the same ODE have a comm...
 CHAPTER 1.3: What is a homogeneous linear ODE? A nonhomogeneous linear ODE? Why ...
 CHAPTER 1.1.1: Solve the ODE by integration. y' cosh 4x
 CHAPTER 1.1.2: Graph a direction field (by a CAS or by hand). In the field graph a...
 CHAPTER 1.1.3: Find a general solution. Show the steps of derivation. Check your a...
 CHAPTER 1.1.4: Test for exactness. If exact, solve. If not, use an integrating fac...
 CHAPTER 1.1.5: Find the general solution. If an initial condition is given, find a...
 CHAPTER 1.1.6: Sketch or graph some of the given curves. Guess what their orthogon...
 CHAPTER 1.1.7: (Change of initial condition) What happens in Prob. 2 if you replac...
 CHAPTER 1.4: What do you know about direction fields and their practical importa...
 CHAPTER 1.1.1: State the order of the ODE. Verify that the given function is a sol...
 CHAPTER 1.1.2: Graph a direction field (by a CAS or by hand). In the field graph a...
 CHAPTER 1.1.3: Find a general solution. Show the steps of derivation. Check your a...
 CHAPTER 1.1.4: Test for exactness. If exact, solve. If not, use an integrating fac...
 CHAPTER 1.1.5: Find the general solution. If an initial condition is given, find a...
 CHAPTER 1.1.6: Sketch or graph some of the given curves. Guess what their orthogon...
 CHAPTER 1.1.7: (Linear ODE) If p and I' in y' + p(x)y = rex) are continuous for al...
 CHAPTER 1.5: Give examples of mechanical problems that lead to ODEs.
 CHAPTER 1.1.1: State the order of the ODE. Verify that the given function is a sol...
 CHAPTER 1.1.2: Graph a direction field (by a CAS or by hand). In the field graph a...
 CHAPTER 1.1.3: Find a general solution. Show the steps of derivation. Check your a...
 CHAPTER 1.1.4: Test for exactness. If exact, solve. If not, use an integrating fac...
 CHAPTER 1.1.5: Find the general solution. If an initial condition is given, find a...
 CHAPTER 1.1.6: Sketch or graph some of the given curves. Guess what their orthogon...
 CHAPTER 1.1.7: (Three possible cases) Find all initial conditions such that lx2  ...
 CHAPTER 1.6: Why do electric circuits lead to ODEs?
 CHAPTER 1.1.1: State the order of the ODE. Verify that the given function is a sol...
 CHAPTER 1.1.2: Graph a direction field (by a CAS or by hand). In the field graph a...
 CHAPTER 1.1.3: Find a general solution. Show the steps of derivation. Check your a...
 CHAPTER 1.1.4: Test for exactness. If exact, solve. If not, use an integrating fac...
 CHAPTER 1.1.5: Find the general solution. If an initial condition is given, find a...
 CHAPTER 1.1.6: Sketch or graph some of the given curves. Guess what their orthogon...
 CHAPTER 1.1.7: (Length of xinterval) In most cases the solution of an initial val...
 CHAPTER 1.7: Make a list of the solution methods considered. Explain each method...
 CHAPTER 1.1.1: State the order of the ODE. Verify that the given function is a sol...
 CHAPTER 1.1.2: Graph a direction field (by a CAS or by hand). In the field graph a...
 CHAPTER 1.1.3: Find a general solution. Show the steps of derivation. Check your a...
 CHAPTER 1.1.4: Test for exactness. If exact, solve. If not, use an integrating fac...
 CHAPTER 1.1.5: Find the general solution. If an initial condition is given, find a...
 CHAPTER 1.1.6: Sketch or graph some of the given curves. Guess what their orthogon...
 CHAPTER 1.1.7: PROJECT. Lipschitz Condition. (A) State the definItion of a Lipschi...
 CHAPTER 1.8: Can certain ODEs be solved by more than one method? Give three exam...
 CHAPTER 1.1.1: State the order of the ODE. Verify that the given function is a sol...
 CHAPTER 1.1.2: Graph a direction field (by a CAS or by hand). In the field graph a...
 CHAPTER 1.1.3: Find a general solution. Show the steps of derivation. Check your a...
 CHAPTER 1.1.4: Test for exactness. If exact, solve. If not, use an integrating fac...
 CHAPTER 1.1.5: Find the general solution. If an initial condition is given, find a...
 CHAPTER 1.1.6: Sketch or graph some of the given curves. Guess what their orthogon...
 CHAPTER 1.1.7: Maximum a) What IS the largest possible a In Example I in the text?
 CHAPTER 1.9: What are integrating factors? Explain the idea. Give examples.
 CHAPTER 1.1.1: Verify that y is a solution of the ODE. Determine from y the partic...
 CHAPTER 1.1.2: Graph a direction field (by a CAS or by hand). In the field graph a...
 CHAPTER 1.1.3: Find the particular solution. Show the steps of derivation, beginni...
 CHAPTER 1.1.4: Test for exactness. If exact, solve. If not, use an integrating fac...
 CHAPTER 1.1.5: Find the general solution. If an initial condition is given, find a...
 CHAPTER 1.1.6: Sketch or graph some of the given curves. Guess what their orthogon...
 CHAPTER 1.1.7: CAS PROJECT. Picard Iteration. (A) Show that by integrating the ODE...
 CHAPTER 1.10: Does every firstorder ODE have a solution? A general solution? Wha...
 CHAPTER 1.1.1: Verify that y is a solution of the ODE. Determine from y the partic...
 CHAPTER 1.1.2: Direction fields are very useful because you can see solutions (as ...
 CHAPTER 1.1.3: Find the particular solution. Show the steps of derivation, beginni...
 CHAPTER 1.1.4: Test for exactness. If exact, solve. If not, use an integrating fac...
 CHAPTER 1.1.5: Find the general solution. If an initial condition is given, find a...
 CHAPTER 1.1.6: Sketch or graph some of the given curves. Guess what their orthogon...
 CHAPTER 1.11: Graph a direction field (by a CAS or by hand) and sketch some of th...
 CHAPTER 1.1.1: Verify that y is a solution of the ODE. Determine from y the partic...
 CHAPTER 1.1.2: Direction fields are very useful because you can see solutions (as ...
 CHAPTER 1.1.3: Find the particular solution. Show the steps of derivation, beginni...
 CHAPTER 1.1.4: Test for exactness. If exact, solve. If not, use an integrating fac...
 CHAPTER 1.1.5: Find the general solution. If an initial condition is given, find a...
 CHAPTER 1.1.6: Sketch or graph some of the given curves. Guess what their orthogon...
 CHAPTER 1.12: Graph a direction field (by a CAS or by hand) and sketch some of th...
 CHAPTER 1.1.1: Verify that y is a solution of the ODE. Determine from y the partic...
 CHAPTER 1.1.2: Direction fields are very useful because you can see solutions (as ...
 CHAPTER 1.1.3: Find the particular solution. Show the steps of derivation, beginni...
 CHAPTER 1.1.4: Test for exactness. If exact, solve. If not, use an integrating fac...
 CHAPTER 1.1.5: Find the general solution. If an initial condition is given, find a...
 CHAPTER 1.1.6: (y as independent variable) Show that (3) may be written dx/dy = f...
 CHAPTER 1.13: Graph a direction field (by a CAS or by hand) and sketch some of th...
 CHAPTER 1.1.1: Verify that y is a solution of the ODE. Determine from y the partic...
 CHAPTER 1.1.2: Direction fields are very useful because you can see solutions (as ...
 CHAPTER 1.1.3: Find the particular solution. Show the steps of derivation, beginni...
 CHAPTER 1.1.4: Test for exactness. If exact, solve. If not, use an integrating fac...
 CHAPTER 1.1.5: Find the general solution. If an initial condition is given, find a...
 CHAPTER 1.1.6: (Family g(x,y) = c) Show that if a family is given asg(x, y) = c, t...
 CHAPTER 1.14: Graph a direction field (by a CAS or by hand) and sketch some of th...
 CHAPTER 1.1.1: (Existence) (A) Does the ODE y'2 = ] have a (real) solution? (B) D...
 CHAPTER 1.1.2: Direction fields are very useful because you can see solutions (as ...
 CHAPTER 1.1.3: Find the particular solution. Show the steps of derivation, beginni...
 CHAPTER 1.1.4: Test for exactness. If exact, solve. If not, use an integrating fac...
 CHAPTER 1.1.5: Find the general solution. If an initial condition is given, find a...
 CHAPTER 1.1.6: (CauchyRiemann equations) Show that for a familyu(x, y) = c = cons...
 CHAPTER 1.15: Find the general solution. Indicate which method in this chapter yo...
 CHAPTER 1.1.1: (Singular solution) An ODE may sometimes have an additional solutio...
 CHAPTER 1.1.2: A body moves on a straight line, with velocity as given. and yet) i...
 CHAPTER 1.1.3: Find the particular solution. Show the steps of derivation, beginni...
 CHAPTER 1.1.4: Test for exactness. If exact, solve. If not, use an integrating fac...
 CHAPTER 1.1.5: Find the general solution. If an initial condition is given, find a...
 CHAPTER 1.1.6: (Fluid flow) Suppose that the streamlines of the flow lpaths of the...
 CHAPTER 1.16: Find the general solution. Indicate which method in this chapter yo...
 CHAPTER 1.1.1: The following problems will give you a first impression of modeling...
 CHAPTER 1.1.2: A body moves on a straight line, with velocity as given. and yet) i...
 CHAPTER 1.1.3: Find the particular solution. Show the steps of derivation, beginni...
 CHAPTER 1.1.4: Test for exactness. If exact, solve. If not, use an integrating fac...
 CHAPTER 1.1.5: Find the general solution. If an initial condition is given, find a...
 CHAPTER 1.1.6: (Electric field) Let the electric equipotential lines (curves of co...
 CHAPTER 1.17: Find the general solution. Indicate which method in this chapter yo...
 CHAPTER 1.1.1: (Falling body) If in Prob. 17 the stone starts at t = 0 from initia...
 CHAPTER 1.1.2: A body moves on a straight line, with velocity as given. and yet) i...
 CHAPTER 1.1.3: Find the particular solution. Show the steps of derivation, beginni...
 CHAPTER 1.1.4: Test for exactness. If exact, solve. If not, use an integrating fac...
 CHAPTER 1.1.5: Using a method of this section or separating variables, find the ge...
 CHAPTER 1.1.6: (Electric field) The lines of electric force of two opposite charge...
 CHAPTER 1.18: Find the general solution. Indicate which method in this chapter yo...
 CHAPTER 1.1.1: (Airplane takeoff) If an airplane has a run of 3 km, statts with a ...
 CHAPTER 1.1.2: (Skydiver) Two forces act on a parachutist, the attraction by the e...
 CHAPTER 1.1.3: Find the particular solution. Show the steps of derivation, beginni...
 CHAPTER 1.1.4: Test for exactness. If exact, solve. If not, use an integrating fac...
 CHAPTER 1.1.5: Using a method of this section or separating variables, find the ge...
 CHAPTER 1.1.6: (Temperature field) Let the isotherms (curves of constant temperatu...
 CHAPTER 1.19: Find the general solution. Indicate which method in this chapter yo...
 CHAPTER 1.1.1: (Subsonic flight) The efficiency of the engines of subsonic airplan...
 CHAPTER 1.1.2: CAS PROJECT. Direction Fields. Discuss direction fields as follows....
 CHAPTER 1.1.3: (Particulal' solution) Introduce limits of integration in (3) such ...
 CHAPTER 1.1.4: Test for exactness. If exact, solve. If not, use an integrating fac...
 CHAPTER 1.1.5: Using a method of this section or separating variables, find the ge...
 CHAPTER 1.1.6: TEAM PROJECT. Conic Sections. (A) State the main steps of the prese...
 CHAPTER 1.20: Find the general solution. Indicate which method in this chapter yo...
 CHAPTER 1.1.1: (Halflife) The halflife of a radioactive substance is the time in...
 CHAPTER 1.1.3: (Curves) Find all curves in the xyplane whose tangents all pass th...
 CHAPTER 1.1.4: Under what conditions for the constants A, B. C, D is (Ax + By) dx ...
 CHAPTER 1.1.5: Using a method of this section or separating variables, find the ge...
 CHAPTER 1.21: Find the general solution. Indicate which method in this chapter yo...
 CHAPTER 1.1.1: (Interest rates) Show by algebra that the investment y(t) from a de...
 CHAPTER 1.1.3: (Cm'ves) Show that any (nonverticaD straight line through the origi...
 CHAPTER 1.1.4: Graph paI1icuiar solutions of the following ODE.proceeding as expla...
 CHAPTER 1.1.5: Using a method of this section or separating variables, find the ge...
 CHAPTER 1.22: Find the general solution. Indicate which method in this chapter yo...
 CHAPTER 1.1.3: (Exponential growth) If the growth rate of the amount of yeast at a...
 CHAPTER 1.1.4: WRITING PROJECT. Working Backward. Start from solutions u(x, v) = c...
 CHAPTER 1.1.5: Using a method of this section or separating variables, find the ge...
 CHAPTER 1.23: Find the general solution. Indicate which method in this chapter yo...
 CHAPTER 1.1.3: (Population model) If in a population of bacteria the birth rate an...
 CHAPTER 1.1.4: TEAM PROJECT. Solution by Several Methods. Show this as indicated. ...
 CHAPTER 1.1.5: Using a method of this section or separating variables, find the ge...
 CHAPTER 1.24: Find the general solution. Indicate which method in this chapter yo...
 CHAPTER 1.1.3: (Radiocal'bon dating) If a fossilized tree is claimed to be 4000 ye...
 CHAPTER 1.1.5: (Investment programs) Bill opens a retirement savings account with ...
 CHAPTER 1.25: Find the general solution. Indicate which method in this chapter yo...
 CHAPTER 1.1.3: (Gompel'tz gl'Owth in tumors) TIle Gompertz model is r' = Av In \'...
 CHAPTER 1.1.5: (Mixing problem) A tank (as in Fig. 9 in Sec. 1.3) contains 1000 ga...
 CHAPTER 1.26: Find the general solution. Indicate which method in this chapter yo...
 CHAPTER 1.1.3: (Dlyel') If wet laundry loses half of its moisture during the firs...
 CHAPTER 1.1.5: (Lake Erie) Lake Erie has a water volume of about 450 km3 and a flo...
 CHAPTER 1.27: Solve the following initial value problems. Indicate the method use...
 CHAPTER 1.1.3: (Alibi?) Jack, arrested when leaving a bar, claims that he has been...
 CHAPTER 1.1.5: (Heating and cooling of a building) Heating and cooling of a buildi...
 CHAPTER 1.28: Solve the following initial value problems. Indicate the method use...
 CHAPTER 1.1.3: (Law of cooling) A thermometer, reading 10C, is brought into a room...
 CHAPTER 1.1.5: (Drug injection) Find and solve the model for drug injection into t...
 CHAPTER 1.29: Solve the following initial value problems. Indicate the method use...
 CHAPTER 1.1.3: (TonicelIi's law) How does the answer in Example 5 (the time when t...
 CHAPTER 1.1.5: (Epidemics) A model for the spread of contagious diseases is obtain...
 CHAPTER 1.30: Solve the following initial value problems. Indicate the method use...
 CHAPTER 1.1.3: (TolTicelli's law) Show that (7) looks reasonable inasmuch as V2gh(...
 CHAPTER 1.1.5: (Extinction vs. unlimited growth) If in a population y(t) the death...
 CHAPTER 1.31: Solve the following initial value problems. Indicate the method use...
 CHAPTER 1.1.3: (Rope) To tie a boat in a harbor. how many times must a rope be wou...
 CHAPTER 1.1.5: (Harvesting renewable resources. Fishing) Suppose that the populati...
 CHAPTER 1.32: Solve the following initial value problems. Indicate the method use...
 CHAPTER 1.1.3: (Mixing) A tank contains 800 gal of water in which 200 Ib of salt i...
 CHAPTER 1.1.5: (Harvesting) In Prob. 32 find and graph the solution satisfying yeO...
 CHAPTER 1.33: ~Heat flow) If the isothelms in a region are x 2  )'2 = c, what ar...
 CHAPTER 1.1.3: WRITING PROJECT. Exponential Increase, Decay, Approach. Collect. or...
 CHAPTER 1.1.5: (Intermittent harvesting) In Prob. 32 assume that you fish for 3 ye...
 CHAPTER 1.34: (Law of cooling) A thennometer showing WaC is brought into a room w...
 CHAPTER 1.1.3: CAS EXPERIMENT. Graphing Solutions. A CAS can usually graph solutio...
 CHAPTER 1.1.5: (Harvesting) If a population of mice (in multiples of 1000) follows...
 CHAPTER 1.35: (Halflife) If 10o/c of a radioactive substance disintegrates in 4 ...
 CHAPTER 1.1.3: TEAM PROJECT. Tonicelli's Law. Suppose that the tank in Example 5 i...
 CHAPTER 1.1.5: (Harvesting) Do you save work in Prob. 34 if you first transform th...
 CHAPTER 1.36: (HaIflife) What is the halflife of a substance if after 5 days, 0...
 CHAPTER 1.1.5: The sum YI + Y2 of two solutions YI and Y2 of the homogeneous equat...
 CHAPTER 1.37: (HaIflife) When will 99% of the substance in Prob. 35 have disinte...
 CHAPTER 1.1.5: Y = 0 (that is, .v(x) = 0 for all x, also written y(x) "'" 0) is a ...
 CHAPTER 1.38: (Air circulation) In a room containing 20000 ft3 of air, 600 ft3 of...
 CHAPTER 1.1.5: The sum of a solution of (I) and a solution of (2) is a solution of...
 CHAPTER 1.39: (Electric field) If the equipotential lines in a region of the x)"...
 CHAPTER 1.1.5: The difference of two solutions of (l) is a solution of (2).
 CHAPTER 1.40: (Chemistry) In a bimolecular reaction A + B ? M, a moles per liter...
 CHAPTER 1.1.5: If Yl is a sulution of (I), what can you say about eYl?
 CHAPTER 1.41: (Population) Find the population y(1) if the birth rate is proporti...
 CHAPTER 1.1.5: If YI and Y2 are solutions of y~ + PYI = rl and Y~ + PY2 = r2, resp...
 CHAPTER 1.42: (Curves) Find all curve~ in the first quadrant of the Ayplane such ...
 CHAPTER 1.1.5: CAS EXPERIMENT. (a) Solve the ODE y'  ylx = x1 cos (l/x). Find a...
 CHAPTER 1.43: (Optics) Lambert's law of absorption9 states that the absorption of...
 CHAPTER 1.1.5: TEAM PROJECT. Riccati Equation, Clairaut Equation. A Riccati equati...
 CHAPTER 1.1.5: (Variation of parameter) Another method of obtaining (4) results fr...
 CHAPTER 1.1.5: TEAM PROJECT. Transformations of ODEs. We have transformed ODEs to ...
Solutions for Chapter CHAPTER 1: Advanced Engineering Mathematics 9th Edition
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter CHAPTER 1
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 220 problems in chapter CHAPTER 1 have been answered, more than 44116 students have viewed full stepbystep solutions from this chapter. Chapter CHAPTER 1 includes 220 full stepbystep solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).