Deck 12: Second-Order Differential Equations

Question 1

Find the general solution of the differential equation $y^{\prime \prime}+5 y^{\prime}+6 y=0$ .

A) $y=k_{1} e^{-5 x}+k_{2} e^{-x}$
B) $y=k_{1} e^{-3 x}+k_{2} e^{-2 x}$
C) $y=k_{1} e^{3 x}+k_{2} e^{2 x}$
D) $y=k_{1} e^{5 x}+k_{2} e^{x}$

Answer 1

$y=k_{1} e^{-3 x}+k_{2} e^{-2 x}$

Question 2

Find the general solution of the differential equation $y^{\prime \prime}+8 y^{\prime}+16 y=0$ .

A) $y=k_{1} e^{-4 x}+k_{2} e^{-4 x}$
B) $y=k_{1} e^{4 x}+k_{2} e^{4 x}$
C) $y=k_{1} e^{-4 x}+k_{2} e^{4 x}$
D) $y=k_{1} e^{-4 x}+k_{2} x e^{-4 x}$

Answer 2

$y=k_{1} e^{-4 x}+k_{2} x e^{-4 x}$

Question 3

Find the general solution of the differential equation $y^{\prime \prime}+6 y^{\prime}+25 y=0$ .

A) $y=e^{4 x}\left(k_{1} \sin 3 x+k_{2} \cos 3 x\right)$
B) $y=e^{-3 x}\left(k_{1} \sin 4 x+k_{2} \cos 4 x\right)$
C) $y=e^{3 x}\left(k_{1} \sin 4 x+k_{2} \cos 4 x\right)$
D) $y=e^{-4 x}\left(k_{1} \sin 3 x+k_{2} \cos 3 x\right)$

Answer 3

$y=e^{-3 x}\left(k_{1} \sin 4 x+k_{2} \cos 4 x\right)$

Question 4

Find the particular solution of the differential equation $y^{\prime \prime}-25 y=0$ if $y=5$ and $y^{\prime}=2$ when $x=0$ .

A) $y=0.4 \sin 5 x+5 \cos 5 x$
B) $y=2.3 e^{5 x}+2.7 e^{-5 x}$
C) $y=2.7 e^{5 x}+2.3 e^{-5 x}$
D) $y=5 \sin 5 x+0.4 \cos 5 x$

Question 5

Find the general solution of the differential equation $y^{\prime \prime}+4 y^{\prime}+3 y=20 \cos x$ .

A) $y=k_{1} e^{-3 x}+k_{2} e^{-x}-4 \sin x+2 \cos x$
B) $y=k_{1} e^{-3 x}+k_{2} e^{-x}+4 \sin x+2 \cos x$
C) $y=k_{1} e^{3 x}+k_{2} e^{x}+4 \sin x+2 \cos x$
D) $y=k_{1} e^{3 x}+k_{2} e^{x}-4 \sin x+2 \cos x$

Question 6

Find the general solution of the differential equation $y^{\prime \prime}-4 y=8 e^{2 x}$ .

A) $y=k_{1} \sin 2 x+k_{2} \cos 2 x+e^{2 x}$
B) $y=k_{1} e^{-2 x}+k_{2} e^{2 x}+2 x e^{2 x}$
C) $y=k_{1} \sin 2 x+k_{2} \cos 2 x+x e^{2 x}$
D) $y=k_{1} e^{-2 x}+k_{2} e^{2 x}+x e^{2 x}$

Question 7

An electric circuit has an inductance $L=0.5 \mathrm{H}$ , a resistance $\mathrm{R}=1000 \Omega$ , and a capacitance $\mathrm{C}=1.0 \times$ $10^{-6} \mathrm{~F}$ . Find the equation for the current $\mathrm{i}$ if the voltage sourœ is $12 \mathrm{~V}$ .

A) $i=e^{1000 t}\left(k_{1} \sin 1000 t+k_{2} \cos 1000 t\right)+0.000012$
B) $i=e^{-1000 t}\left(k_{1} \sin 1000 t+k_{2} \cos 1000 t\right)+0.000012$
C) $i=e^{1000 t}\left(k_{1} \sin 1000 t+k_{2} \cos 1000 t\right)$
D) $i=e^{-1000 t}\left(k_{1} \sin 1000 t+k_{2} \cos 1000 t\right)$

Question 8

Using a table of Laplace transforms, find the Laplace transform of $f(t)=t e^{-3 t}$ .

A) $\frac{1}{(s-3)^{2}}$
B) $\frac{\mathrm{s}}{(\mathrm{s}+3)^{2}}$
C) $\frac{s^{2}}{s-3}$
D) $\frac{1}{(s+3)^{2}}$

Question 9

Using a table of Laplace transforms, find the inverse transform of $F(s)=\frac{1}{s^{2}+10 s+29}$ .

A) $f(t)=\frac{1}{2} e^{-2 t} \sin 5 t$

B) $f(t)=e^{-2 t} \cos 5 t$

C) $f(t)=e^{-2 t} \sin 5 t$

D) $f(t)=\frac{1}{2} e^{-2 t} \cos 5 t$

Question 10

For the problems below, solve each differential equation.

- $y^{\prime \prime}+4 y^{\prime}-21 y=0$

Question 11

For the problems below, solve each differential equation.

- $y^{\prime \prime}+8 y^{\prime}=0$

Question 12

Find the particular solution subject to the given conditions. $\mathrm{y}^{\prime \prime}-4 \mathrm{y}^{\prime}-12 \mathrm{y}=0 ; \mathrm{y}=1$ and $\mathrm{y}^{\prime}=-18$ when $\mathrm{x}=0$ .

Question 13

For the problems below, solve each differential equation.

- $y^{\prime \prime}-22 y^{\prime}+121 y=0$

Question 14

For the problems below, solve each differential equation.

- $y^{\prime \prime}-4 y^{\prime}+29 y=0$

Question 15

Find the particular solution subject to the given conditions. $$\mathrm{y}^{\prime \prime}+81 \mathrm{y}=0 ; \mathrm{y}=7 \text { and } \mathrm{y}^{\prime}=-27 \text { when } \mathrm{x}=\frac{\pi}{18}$$

Question 16

For the problems below, solve each differential equation.

- $y^{\prime \prime}+2 y^{\prime}-15 y=x+e^{2 x}$

Question 17

For the problems below, solve each differential equation.

- $\mathrm{y}^{\prime \prime}+\mathrm{y}=4 \sin \mathrm{x}$

Question 18

A spring is stretched $6 \mathrm{in}$ . by a weight of $10 \mathrm{lb}$ . If the weight is displaced a distance of $4 \mathrm{in}$ . from a rest position and then released, find the equation of motion.

Question 19

A spring is stretched $8 \mathrm{in}$ . by a weight of $25 \mathrm{lb}$ . A damping force exerts a force of $6 \mathrm{lb}$ for a velocity of $5 \mathrm{in}$ ./s. If the weight is displaced from the rest position and then released, find the general equation of motion.

Question 20

An electric circuit has an inductance $L=0.4 \mathrm{H}$ , a resistance $R=250 \Omega$ , and a capacitance $C$ $=5 \times 10^{-4}$ F. Find the equation for the current $i$ . (Hint: $\frac{d^{2} i}{d t^{2}}+\frac{R d i}{L d t}+\frac{1}{C L} i=0$ )

Question 21

Use a table of Laplace Transforms to find the Laplace transform of each function $\mathrm{f}(\mathrm{t})$ in the problems below.

- $f(t)=e^{4 t} \cos 8 t+5 t e^{10 t}$

Question 22

Use a table of Laplace Transforms to find the Laplace transform of each function $\mathrm{f}(\mathrm{t})$ in the problems below.

- $f(t)=\frac{t^{4}}{24}+1-\cos 6 t$

Question 23

Find the inverse transform for the function $F(s)$ . $F(s)=\frac{9}{s^{2}+81}+\frac{1}{(s+5)^{3}}$

Question 24

For the problems below, solve each differential equation subject to the given conditions by using Laplace transforms.

- $y^{\prime \prime}-5 y^{\prime}-24 y=0 ; y(0)=17$ and $y^{\prime}(0)=26$

Question 25

For the problems below, solve each differential equation subject to the given conditions by using Laplace transforms.

- $y^{\prime \prime}-8 y^{\prime}+16 y=t e^{4 t} ; y(0)=0$ and $y^{\prime}(0)=0$