 1.5.1: Find general solutions of the differential equations in 1 through 2...
 1.5.2: Find general solutions of the differential equations in 1 through 2...
 1.5.3: Find general solutions of the differential equations in 1 through 2...
 1.5.4: Find general solutions of the differential equations in 1 through 2...
 1.5.5: Find general solutions of the differential equations in 1 through 2...
 1.5.6: Find general solutions of the differential equations in 1 through 2...
 1.5.7: Find general solutions of the differential equations in 1 through 2...
 1.5.8: Find general solutions of the differential equations in 1 through 2...
 1.5.9: Find general solutions of the differential equations in 1 through 2...
 1.5.10: Find general solutions of the differential equations in 1 through 2...
 1.5.11: Find general solutions of the differential equations in 1 through 2...
 1.5.12: Find general solutions of the differential equations in 1 through 2...
 1.5.13: Find general solutions of the differential equations in 1 through 2...
 1.5.14: Find general solutions of the differential equations in 1 through 2...
 1.5.15: Find general solutions of the differential equations in 1 through 2...
 1.5.16: Find general solutions of the differential equations in 1 through 2...
 1.5.17: Find general solutions of the differential equations in 1 through 2...
 1.5.18: Find general solutions of the differential equations in 1 through 2...
 1.5.19: Find general solutions of the differential equations in 1 through 2...
 1.5.20: Find general solutions of the differential equations in 1 through 2...
 1.5.21: Find general solutions of the differential equations in 1 through 2...
 1.5.22: Find general solutions of the differential equations in 1 through 2...
 1.5.23: Find general solutions of the differential equations in 1 through 2...
 1.5.24: Find general solutions of the differential equations in 1 through 2...
 1.5.25: Find general solutions of the differential equations in 1 through 2...
 1.5.26: Solve the differential equations in 26 through 28 by regarding y as...
 1.5.27: Solve the differential equations in 26 through 28 by regarding y as...
 1.5.28: Solve the differential equations in 26 through 28 by regarding y as...
 1.5.29: Express the general solution of dy=dx D 1 C 2xy in terms of the err...
 1.5.30: Express the solution of the initial value problem 2x dy dx D y C 2x...
 1.5.31: (a) Show that yc.x/ D CeR P .x/ dx is a general solution of dy=dx C...
 1.5.32: (a) Find constants A and B such that yp.x/ D A sin x C B cos x is a...
 1.5.33: A tank contains 1000 liters (L) of a solution consisting of 100 kg ...
 1.5.34: Consider a reservoir with a volume of 8 billion cubic feet (ft3) an...
 1.5.35: Consider a reservoir with a volume of 8 billion cubic feet (ft3) an...
 1.5.36: A tank initially contains 60 gal of pure water. Brine containing 1 ...
 1.5.37: A 400gal tank initially contains 100 gal of brine containing 50 lb...
 1.5.38: Consider the cascade of two tanks shown in Fig. 1.5.5, with V1 D 10...
 1.5.39: Suppose that in the cascade shown in Fig. 1.5.5, tank 1 initially c...
 1.5.40: A multiple cascade is shown in Fig. 1.5.6. At time t D 0, tank 0 co...
 1.5.41: A 30yearold woman accepts an engineering position with a starting...
 1.5.42: Suppose that a falling hailstone with density D 1 starts from rest ...
 1.5.43: Suppose that a falling hailstone with density D 1 starts from rest ...
 1.5.44: Figure 1.5.8 shows a slope field and typical solution curves for th...
 1.5.45: The incoming water has a pollutant concentration of c.t / D 10 lite...
 1.5.46: The incoming water has pollutant concentration c.t / D 10.1 C cost ...
Solutions for Chapter 1.5: Linear FirstOrder Equations
Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling  5th Edition
ISBN: 9780321796981
Solutions for Chapter 1.5: Linear FirstOrder Equations
Get Full SolutionsDifferential Equations and Boundary Value Problems: Computing and Modeling was written by and is associated to the ISBN: 9780321796981. This expansive textbook survival guide covers the following chapters and their solutions. Since 46 problems in chapter 1.5: Linear FirstOrder Equations have been answered, more than 16611 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations and Boundary Value Problems: Computing and Modeling, edition: 5. Chapter 1.5: Linear FirstOrder Equations includes 46 full stepbystep solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Column space C (A) =
space of all combinations of the columns of A.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Outer product uv T
= column times row = rank one matrix.

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).

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).