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Calculus

James Stewart

Chapter 17

Second-Order Differential Equations - all with Video Answers

Educators

Section 1

17.1 Exercises

02:33

Problem 1

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

Bryan Lynn
Bryan Lynn
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01:00

Problem 2

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

Raj Bala
Raj Bala
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02:28

Problem 3

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

Linh Vu
Linh Vu
Numerade Educator
01:35

Problem 4

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

Linh Vu
Linh Vu
Numerade Educator
02:00

Problem 5

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

Linh Vu
Linh Vu
Numerade Educator
02:06

Problem 6

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

Linh Vu
Linh Vu
Numerade Educator
01:52

Problem 7

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

Linh Vu
Linh Vu
Numerade Educator
01:00

Problem 8

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

Raj Bala
Raj Bala
Numerade Educator
02:21

Problem 9

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

Linh Vu
Linh Vu
Numerade Educator

Problem 10

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

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01:56

Problem 11

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

Pravakar Paul
Pravakar Paul
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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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02:22

Problem 13

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

Linh Vu
Linh Vu
Numerade Educator
03:08

Problem 14

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

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:39

Problem 15

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

Eric Mockensturm
Eric Mockensturm
Numerade Educator
03:08

Problem 16

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

Eric Mockensturm
Eric Mockensturm
Numerade Educator
03:16

Problem 17

Solve the initial-value problem.
$y^{\prime \prime}-6 y^{\prime}+8 y=0, \quad y(0)=2, \quad y^{\prime}(0)=2$

Linh Vu
Linh Vu
Numerade Educator
03:07

Problem 18

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

Linh Vu
Linh Vu
Numerade Educator
03:18

Problem 19

Solve the initial-value problem.
$9 y^{\prime \prime}+12 y^{\prime}+4 y=0, \quad y(0)=1, \quad y^{\prime}(0)=0$

Linh Vu
Linh Vu
Numerade Educator
03:25

Problem 20

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

Linh Vu
Linh Vu
Numerade Educator
04:14

Problem 21

Solve the initial-value problem.
$y^{\prime \prime}-6 y^{\prime}+10 y=0, \quad y(0)=2, \quad y^{\prime}(0)=3$

Linh Vu
Linh Vu
Numerade Educator
02:55

Problem 22

Solve the initial-value problem.
$4 y^{\prime \prime}-20 y^{\prime}+25 y=0, \quad y(0)=2, \quad y^{\prime}(0)=-3$

Linh Vu
Linh Vu
Numerade Educator
04:02

Problem 23

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

Linh Vu
Linh Vu
Numerade Educator
05:12

Problem 24

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

Linh Vu
Linh Vu
Numerade Educator
02:24

Problem 25

Solve the boundary-value problem, if possible.
$y^{\prime \prime}+4 y=0, \quad y(0)=5, \quad y(\pi / 4)=3$

Linh Vu
Linh Vu
Numerade Educator
03:09

Problem 26

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

Linh Vu
Linh Vu
Numerade Educator
02:33

Problem 27

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

Linh Vu
Linh Vu
Numerade Educator
02:39

Problem 28

Solve the boundary-value problem, if possible.
$y^{\prime \prime}-8 y^{\prime}+17 y=0, \quad y(0)=3, \quad y(\pi)=2$

Linh Vu
Linh Vu
Numerade Educator
02:24

Problem 29

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

Linh Vu
Linh Vu
Numerade Educator
02:17

Problem 30

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

Linh Vu
Linh Vu
Numerade Educator
02:42

Problem 31

Solve the boundary-value problem, if possible.
$y^{\prime \prime}+4 y^{\prime}+20 y=0, \quad y(0)=1, \quad y(\pi)=2$

Linh Vu
Linh Vu
Numerade Educator
02:47

Problem 32

Solve the boundary-value problem, if possible.
$y^{\prime \prime}+4 y^{\prime}+20 y=0, \quad y(0)=1, \quad y(\pi)=e^{-2 \pi}$

Linh Vu
Linh Vu
Numerade Educator
04:54

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.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
04:22

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 $\lim _{x \rightarrow \infty} y(x)=0$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
06:19

Problem 35

Consider the boundary-value problem $y^{\prime \prime}-2 y^{\prime}+2 y=0$, $y(a)=c, y(b)=d$.
(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?

Eric Mockensturm
Eric Mockensturm
Numerade Educator