 2.1: In each of 1 through 32, solve the given differential equation. If ...
 2.2: In each of 1 through 32, solve the given differential equation. If ...
 2.3: In each of 1 through 32, solve the given differential equation. If ...
 2.4: In each of 1 through 32, solve the given differential equation. If ...
 2.5: In each of 1 through 32, solve the given differential equation. If ...
 2.6: In each of 1 through 32, solve the given differential equation. If ...
 2.7: In each of 1 through 32, solve the given differential equation. If ...
 2.8: In each of 1 through 32, solve the given differential equation. If ...
 2.9: In each of 1 through 32, solve the given differential equation. If ...
 2.10: In each of 1 through 32, solve the given differential equation. If ...
 2.11: In each of 1 through 32, solve the given differential equation. If ...
 2.12: In each of 1 through 32, solve the given differential equation. If ...
 2.13: In each of 1 through 32, solve the given differential equation. If ...
 2.14: In each of 1 through 32, solve the given differential equation. If ...
 2.15: In each of 1 through 32, solve the given differential equation. If ...
 2.16: In each of 1 through 32, solve the given differential equation. If ...
 2.17: In each of 1 through 32, solve the given differential equation. If ...
 2.18: In each of 1 through 32, solve the given differential equation. If ...
 2.19: In each of 1 through 32, solve the given differential equation. If ...
 2.20: In each of 1 through 32, solve the given differential equation. If ...
 2.21: In each of 1 through 32, solve the given differential equation. If ...
 2.22: In each of 1 through 32, solve the given differential equation. If ...
 2.23: In each of 1 through 32, solve the given differential equation. If ...
 2.24: In each of 1 through 32, solve the given differential equation. If ...
 2.25: In each of 1 through 32, solve the given differential equation. If ...
 2.26: In each of 1 through 32, solve the given differential equation. If ...
 2.27: In each of 1 through 32, solve the given differential equation. If ...
 2.28: In each of 1 through 32, solve the given differential equation. If ...
 2.29: In each of 1 through 32, solve the given differential equation. If ...
 2.30: In each of 1 through 32, solve the given differential equation. If ...
 2.31: In each of 1 through 32, solve the given differential equation. If ...
 2.32: In each of 1 through 32, solve the given differential equation. If ...
 2.33: Riccati Equations. The equation dy dt = q1(t) + q2(t)y + q3(t)y2 is...
 2.34: Using the method of and the given particular solution, solve each o...
 2.35: The propagation of a single action in a large population (for examp...
 2.36: Equations with the Dependent Variable Missing. For a second order d...
 2.37: Equations with the Dependent Variable Missing. For a second order d...
 2.38: Equations with the Dependent Variable Missing. For a second order d...
 2.39: Equations with the Dependent Variable Missing. For a second order d...
 2.40: Equations with the Dependent Variable Missing. For a second order d...
 2.41: Equations with the Dependent Variable Missing. For a second order d...
 2.42: Equations with the Independent Variable Missing. Consider second or...
 2.43: Equations with the Independent Variable Missing. Consider second or...
 2.44: Equations with the Independent Variable Missing. Consider second or...
 2.45: Equations with the Independent Variable Missing. Consider second or...
 2.46: Equations with the Independent Variable Missing. Consider second or...
 2.47: Equations with the Independent Variable Missing. Consider second or...
 2.48: In each of 48 through 51, solve the given initial value problem usi...
 2.49: In each of 48 through 51, solve the given initial value problem usi...
 2.50: In each of 48 through 51, solve the given initial value problem usi...
 2.51: In each of 48 through 51, solve the given initial value problem usi...
Solutions for Chapter 2: First Order Differential Equations
Full solutions for Elementary Differential Equations and Boundary Value Problems  10th Edition
ISBN: 9780470458310
Solutions for Chapter 2: First Order Differential Equations
Get Full SolutionsSince 51 problems in chapter 2: First Order Differential Equations have been answered, more than 17849 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310. Chapter 2: First Order Differential Equations includes 51 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).

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

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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

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

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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