Chapter 17

SECOND-ORDER DIFFERENTIAL EQUATIONS

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MW

Problem 1

Solve the differential equation.
$y^{\prime \prime}-y^{\prime}-6 y=0$

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Problem 2

Solve the differential equation.
$y^{\prime \prime}+4 y^{\prime}+4 y=0$

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Problem 3

Solve the differential equation.
$y^{\prime \prime}+16 y=0$

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Problem 4

Solve the differential equation.
$y^{\prime \prime}-8 y^{\prime}+12 y=0$

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Problem 5

Solve the differential equation.
$9 y^{\prime \prime}-12 y^{\prime}+4 y=0$

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Problem 6

Solve the differential equation.
$25 y^{\prime \prime}+9 y=0$

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Problem 7

Solve the differential equation.
$y=2 y^{\prime \prime}$

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Problem 8

Solve the differential equation.
$y^{\prime \prime}-4 y^{\prime}+y=0$

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Problem 9

Solve the differential equation.
$y^{\prime \prime}-4 y^{\prime}+13 y=0$

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Problem 10

Solve the differential equation.
$y^{\prime \prime}+3 y^{\prime}=0$

MW
Madeleine W.
Numerade Educator

Problem 11

Solve the differential equation.
$2 \frac{d^{2} y}{d t^{2}}+2 \frac{d y}{d t}-y=0$

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Problem 12

Solve the differential equation.
$8 \frac{d^{2} y}{d t^{2}}+12 \frac{d y}{d t}+5 y=0$

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Problem 13

Solve the differential equation.
$100 \frac{d^{2} P}{d t^{2}}+200 \frac{d P}{d t}+101 P=0$

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Problem 14

Graph the two basic solutions of the differential equation and several other solutions. What features do the solutions have in common?
$\frac{d^{2} y}{d x^{2}}+4 \frac{d y}{d x}+20 y=0$

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Problem 15

Graph the two basic solutions of the differential equation and several other solutions. What features do the solutions have in common?
$5 \frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}-3 y=0$

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Problem 16

Graph the two basic solutions of the differential equation and several other solutions. What features do the solutions have in common?
$9 \frac{d^{2} y}{d x^{2}}+6 \frac{d y}{d x}+y=0$

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Problem 17

Solve the initial-value problem.
$2 y^{\prime \prime}+5 y^{\prime}+3 y=0, \quad y(0)=3, \quad y(0)=-4$

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Problem 18

Solve the initial-value problem.
$y^{\prime \prime}+3 y=0, \quad y(0)=1, \quad y^{\prime}(0)=3$

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Problem 19

Solve the initial-value problem.
$4 y^{\prime \prime}-4 y^{\prime}+y=0, \quad y(0)=1, \quad y(0)=-1.5$

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Problem 20

Solve the initial-value problem.
$2 y^{\prime \prime}+5 y^{\prime}-3 y=0, \quad y(0)=1, \quad y(0)=4$

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Problem 21

Solve the initial-value problem.
$y^{\prime \prime}+16 y=0, \quad y(\pi / 4)=-3, \quad y(\pi / 4)=4$

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Problem 22

Solve the initial-value problem.
$y^{\prime \prime}-2 y^{\prime}+5 y=0, \quad y(\pi)=0, \quad y^{\prime}(\pi)=2$

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Problem 23

Solve the initial-value problem.
$y^{\prime \prime}+2 y^{\prime}+2 y=0, \quad y(0)=2, \quad y(0)=1$

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Problem 24

Solve the initial-value problem.
$y^{\prime \prime}+12 y^{\prime}+36 y=0, \quad y(1)=0, \quad y^{\prime}(1)=1$

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Problem 25

Solve the boundary-value problem, if possible.
$4 y^{\prime \prime}+y=0, \quad y(0)=3, \quad y(\pi)=-4$

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Problem 26

Solve the boundary-value problem, if possible.
$y^{\prime \prime}+2 y^{\prime}=0, \quad y(0)=1, \quad y(1)=2$

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Problem 27

Solve the boundary-value problem, if possible.
$y^{\prime \prime}-3 y^{\prime}+2 y=0, \quad y(0)=1, \quad y(3)=0$

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Problem 28

Solve the boundary-value problem, if possible.
$y^{\prime \prime}+100 y=0, \quad y(0)=2, \quad y(\pi)=5$

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Problem 29

Solve the boundary-value problem, if possible.
$y^{\prime \prime}-6 y^{\prime}+25 y=0, \quad y(0)=1, \quad y(\pi)=2$

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Problem 30

Solve the boundary-value problem, if possible.
$y^{\prime \prime}-6 y^{\prime}+9 y=0, \quad y(0)=1, \quad y(1)=0$

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Problem 31

Solve the boundary-value problem, if possible.
$y^{\prime \prime}+4 y^{\prime}+13 y=0, \quad y(0)=2, \quad y(\pi / 2)=1$

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Problem 32

Solve the boundary-value problem, if possible.
$9 y^{\prime \prime}-18 y^{\prime}+10 y=0, \quad y(0)=0, \quad y(\pi)=1$

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Problem 33

Let $L$ be a nonzero real number.
(a) Show that the boundary-value problem $y^{\prime \prime}+\lambda y=0$ , $y(0)=0, y(L)=0$ 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 $a y^{\prime \prime}+b y^{\prime}+c y=0,$ show that $\lim _{x \rightarrow \infty} y(x)=0$

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