Fundamentals of differential equations 9th edition slader

1. Second-order; linear. 2. Third-order; nonlinear because of (dy/dx) 4. 3. The differential equation is first-order. Writing it in the form x(dy/dx) + y 2 = 1, we see that it is nonlinear in y because of y 2. However, writing it in the form (y 2 − 1)(dx/dy) + x = 0, we see that it is linear in x. 4. The differential equation is first-order. Writing it in the form u(dv/du) + (1 + u)v = ue u we see that it is linear in v. However, writing it in the form (v + uv − ue u)(du/dv) + u = 0, we see that it is nonlinear in u. 5. Fourth-order; linear 6. Second-order; nonlinear because of cos(r + u) 7. Second-order; nonlinear because of 1 + (dy/dx) 2 8. Second-order; nonlinear because of 1/R 2 9. Third-order; linear 10. Second-order; nonlinear because of ˙ x 2 11. From y = e −x/2 we obtain y = − 1 2 e −x/2. Then 2y + y = −e −x/2 + e −x/2 = 0. 12. From y = 6 5 − 6 5 e −20t we obtain dy/dt = 24e −20t , so that dy dt + 20y = 24e −20t + 20 6 5 − 6 5 e −20t = 24. 13. From y = e 3x cos 2x we obtain y = 3e 3x cos 2x − 2e 3x sin 2x and y = 5e 3x cos 2x − 12e 3x sin 2x, so that y − 6y + 13y = 0. 14. From y = − cos x ln(sec x + tan x) we obtain y = −1 + sin x ln(sec x + tan x) and y = tan x + cos x ln(sec x + tan x). Then y + y = tan x. 15. Writing ln(2X − 1) − ln(X − 1) = t and differentiating implicitly we obtain 2 2X − 1 dX dt − 1 X − 1 dX dt = 1 2 2X − 1 − 1 X − 1 dX dt = 1 2X − 2 − 2X + 1 (2X − 1)(X − 1) dX dt = 1 dX dt = −(2X − 1)(X − 1) = (X − 1)(1 − 2X). Exponentiating both sides of the implicit solution we obtain 2X − 1 X − 1 = e t =⇒ 2X − 1 = Xe t − e t =⇒ (e t − 1) = (e t − 2)X =⇒ X = e t − 1 e t − 2. Solving e t − 2 = 0 we get t = ln 2. Thus, the solution is defined on (−∞, ln 2) or on (ln 2, ∞). The graph of the solution defined on (−∞, ln 2) is dashed, and the graph of the solution defined on (ln 2, ∞) is solid.

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• Fundamentals of Differential Equations

R. Kent Nagle

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Chapters

1

Introduction

4 sections

84 questions

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2

First-Order Differential Equations

6 sections

184 questions

RB

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3

Mathematical Models and Numerical Methods Involving First-Order Equations

7 sections

117 questions

RB

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4

Linear Second-Order Equations

10 sections

300 questions

WJ

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5

Introduction to Systems and Phase Plane Analysis

8 sections

156 questions

RB

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6

Theory of Higher-Order Linear Differential Equations

4 sections

117 questions

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7

Laplace Transforms

10 sections

320 questions

RB

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8

Series Solutions of Differential Equations

8 sections

255 questions

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9

Matrix Methods for Linear Systems

8 sections

245 questions

WJ

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10

Partial Differential Equations

7 sections

155 questions

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