# Calculus Early Transcendentals

## Educators

Problem 1

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

Check back soon!

Problem 2

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

Check back soon!

Problem 3

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

Check back soon!

Problem 4

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

Check back soon!

Problem 5

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

Check back soon!

Problem 6

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

Check back soon!

Problem 7

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

Check back soon!

Problem 8

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

Check back soon!

Problem 9

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

Check back soon!

Problem 10

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

Check back soon!

Problem 11

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

Check back soon!

Problem 12

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

Check back soon!

Problem 13

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

Check back soon!

Problem 14

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

Check back soon!

Problem 15

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

Check back soon!

Problem 16

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

Check back soon!

Problem 17

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

Check back soon!

Problem 18

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

Check back soon!

Problem 19

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

Check back soon!

Problem 20

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

Check back soon!

Problem 21

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

Check back soon!

Problem 22

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

Check back soon!

Problem 23

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

Check back soon!

Problem 24

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

Check back soon!

Problem 25

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

Check back soon!

Problem 26

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

Check back soon!

Problem 27

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

Check back soon!

Problem 28

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

Check back soon!

Problem 29

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

Check back soon!

Problem 30

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

Check back soon!

Problem 31

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

Check back soon!

Problem 32

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

Check back soon!

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.

Check back soon!

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

Check back soon!

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?

Check back soon!