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Chapter 17

Second-Order Differential Equations

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Problem 1

Solve the differential equation.

$ y'' - y' - 6y = 0 $

PP
Pravakar P.
Numerade Educator

Problem 2

Solve the differential equation.

$ y'' - 6y' + 9y = 0 $

PP
Pravakar P.
Numerade Educator

Problem 3

Solve the differential equation.

$ y'' + 2y = 0 $

PP
Pravakar P.
Numerade Educator

Problem 4

Solve the differential equation.

$ y'' + y' - 12y = 0 $

PP
Pravakar P.
Numerade Educator

Problem 5

Solve the differential equation.

$ 4y'' + 4y' + y = 0 $

PP
Pravakar P.
Numerade Educator

Problem 6

Solve the differential equation.

$ 9y'' + 4y = 0 $

PP
Pravakar P.
Numerade Educator

Problem 7

Solve the differential equation.

$ 3y'' = 4y' $

PP
Pravakar P.
Numerade Educator

Problem 8

Solve the differential equation.

$ y = y'' $

PP
Pravakar P.
Numerade Educator

Problem 9

Solve the differential equation.

$ y'' - 4y' + 13y = 0 $

PP
Pravakar P.
Numerade Educator

Problem 10

Solve the differential equation.

$ 3y'' + 4y' - 3y = 0 $

PP
Pravakar P.
Numerade Educator

Problem 11

Solve the differential equation.

$ 2 \dfrac{d^2y}{dt^2} + 2 \dfrac{dy}{dt} - y = 0 $

PP
Pravakar P.
Numerade Educator

Problem 12

Solve the differential equation.

$ \dfrac{d^2R}{dt^2} + 6 \dfrac{dR}{dt} + 34R = 0 $

PP
Pravakar P.
Numerade Educator

Problem 13

Solve the differential equation.

$ 3 \dfrac{d^2V}{dt^2} + 4 \dfrac{dV}{dt} + 3V = 0 $

PP
Pravakar P.
Numerade Educator

Problem 14

Graph the two basic solutions along with several other solutions of the differential equation. What features do the solutions have in common?

$ 4 \dfrac{d^2y}{dx^2} - 4 \dfrac{dy}{dx} + y = 0 $

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Problem 15

Graph the two basic solutions along with several other solutions of the differential equation. What features do the solutions have in common?

$ \dfrac{d^2y}{dx^2} + 2 \dfrac{dy}{dx} + 2y = 0 $

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Problem 16

Graph the two basic solutions along with several other solutions of the differential equation. What features do the solutions have in common?

$ 2 \dfrac{d^2y}{dx^2} + \dfrac{dy}{dx} - y = 0 $

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Problem 17

Solve the initial-value problem.

$ y" + 3y = 0 $, $ y(0) = 1 $, $ y'(0) = 3 $

PP
Pravakar P.
Numerade Educator

Problem 18

Solve the initial-value problem.

$ y" - 2y' - 3y = 0 $, $ y(0) = 2 $, $ y'(0) = 2 $

PP
Pravakar P.
Numerade Educator

Problem 19

Solve the initial-value problem.

$ 9y" + 12y' + 4y = 0 $, $ y(0) = 1 $, $ y'(0) = 0 $

PP
Pravakar P.
Numerade Educator

Problem 20

Solve the initial-value problem.

$ 3y" - 2y' - y = 0 $, $ y(0) = 0 $, $ y'(0) = -4 $

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Problem 21

Solve the initial-value problem.

$ y" - 6y' + 10y = 0 $, $ y(0) = 2 $, $ y'(0) = 3 $

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Problem 22

Solve the initial-value problem.

$ 4y" - 20y' + 25y = 0 $, $ y(0) = 2 $, $ y'(0) = -3 $

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Problem 23

Solve the initial-value problem.

$ y" - y' - 12y = 0 $, $ y(1) = 0 $, $ y'(1) = 1 $

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Problem 24

Solve the initial-value problem.

$ 4y" + 4y' + 3y = 0 $, $ y(0) = 0 $, $ y'(0) = 1 $

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Problem 25

Solve the boundary-value problem, if possible.

$ y" + 16y = 0 $, $ y(0) = -3 $, $ y(\pi/8) = 2 $

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Problem 26

Solve the boundary-value problem, if possible.

$ y'' + 6y' = 0 $, $ y(0) = 1 $, $ y(1) = 0 $

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Problem 27

Solve the boundary-value problem, if possible.

$ y'' + 4y' + 4y = 0 $, $ y(0) = 2 $, $ y(1) = 0 $

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Problem 28

Solve the boundary-value problem, if possible.

$ y'' - 8y' + 17y = 0 $, $ y(0) = 3 $, $ y(\pi) = 2 $

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Problem 29

Solve the boundary-value problem, if possible.

$ y'' = y' $, $ y(0) = 1 $, $ y(1) = 2 $

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Problem 30

Solve the boundary-value problem, if possible.

$ 4y'' - 4y' + y = 0 $, $ y(0) = 4 $, $ y(2) = 0 $

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Problem 31

Solve the boundary-value problem, if possible.

$ y" + 4y' + 20y = 0 $, $ y(0) = 1 $, $ y(\pi) = 2 $

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Problem 32

Solve the boundary-value problem, if possible.

$ y" + 4y' + 20y = 0 $, $ y(0) = 1 $, $ y(\pi) = e^{-2\pi} $

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Problem 33

Let $ L $ be a nonzero real number.

(a) Show that the boundary-value problem $ y'' + \lambda y = 0 $, $ y(0) = 0 $, $ y(L) = 0 $ has only the trivial solution $ y = 0 $ for the cases $ \lambda = 0 $ and $ \lambda < 0 $.

(b) For the case $ \lambda > 0 $, find the values of $ \lambda $ for which this problem has a nontrivial solution and give the corresponding solution.

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Problem 34

If $ a $, $ b $, and $ c $ are all positive constants and $ y(x) $ is a solution of the differential equation $ ay'' + by' + cy = 0 $, show that $ \lim_{x \to \infty} y(x) = 0 $.

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Problem 35

Consider the boundary-value problem $ y''- 2y' + 2y = 0 $, $ y(a) = c $, $ y(b) = d $.

(a) If this problem has a unique solution, how are $ a $ and $ b $ related?

(b) If this problem has no solution, how are $ a $, $ b $, $ c $, and $ d $ related?

(c) If this problem has infinitely many solutions, how are $ a $, $ b $, $ c $, and $ d $ related?

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