Question 1

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

Accepted Answer

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

Question 2

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 $12 \mathrm{~V}$.

Accepted Answer

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

Question 3

For the problems below, solve each differential equation.
-$y^{\prime \prime}+2 y^{\prime}-15 y=x+e^{2 x}$

Accepted Answer

$y=k_{1} e^{3 x}+k_{2} e^{-5 x}-\frac{1}{15} x-\frac{2}{225}-\frac{1}{7} e^{2 x}$

Question 4

Use a table of Laplace Transforms to find the Laplace transform of each function $\mathrm{f}(\mathrm{t})$ in the problems below.
-  $f(t)=\frac{t^{4}}{24}+1-\cos 6 t$

Accepted Answer

The answer of Use a table of Laplace Transforms to...

Question 5

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 $y^{\prime}(0)=26$

Accepted Answer

The answer of For the problems below, solve each differential...

Question 6

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.

Accepted Answer

The answer of A spring is stretched $6 \mathrm{in}$. by...

Question 7

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 $y^{\prime}(0)=0$

Accepted Answer

The answer of For the problems below, solve each differential...

Question 8

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

Accepted Answer

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

Question 9

For the problems below, solve each differential equation.
- $y^{\prime \prime}-4 y^{\prime}+29 y=0$

Accepted Answer

The answer of For the problems below, solve each differential...

Question 10

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: $\frac{d^{2} i}{d t^{2}}+\frac{R d i}{L d t}+\frac{1}{C L} i=0$ )

Accepted Answer

The answer of An electric circuit has an inductance \(L=0.4...

Question 11

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

Accepted Answer

The answer of Find the particular solution subject to the...

Question 12

For the problems below, solve each differential equation.
-  $\mathrm{y}^{\prime \prime}+\mathrm{y}=4 \sin \mathrm{x}$

Accepted Answer

The answer of For the problems below, solve each differential...

Question 13

For the problems below, solve each differential equation.
- $y^{\prime \prime}+8 y^{\prime}=0$

Accepted Answer

The answer of For the problems below, solve each differential...

Question 14

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

Accepted Answer

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)  $f(t)=\frac{1}{2} e^{-2 t} \cos 5 t$ 
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 15

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

Accepted Answer

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)  $y=k_{1} e^{-2 x}+k_{2} e^{2 x}+x e^{2 x}$ 
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 16

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

Accepted Answer

The answer of Find the inverse transform for the function...

Question 17

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 $\mathrm{x}=0$.

Accepted Answer

The answer of Find the particular solution subject to the...

Question 18

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

Accepted Answer

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)  $y=5 \sin 5 x+0.4 \cos 5 x$ 
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 19

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

Accepted Answer

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)  $y=k_{1} e^{3 x}+k_{2} e^{x}-4 \sin x+2 \cos x$ 
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 20

Use a table of Laplace Transforms to find the Laplace transform of each function $\mathrm{f}(\mathrm{t})$ in the problems below.
-  $f(t)=e^{4 t} \cos 8 t+5 t e^{10 t}$

Accepted Answer

The answer of Use a table of Laplace Transforms to...

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

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 $12 \mathrm{~V}$ .

For the problems below, solve each differential equation. - $y^{\prime \prime}+2 y^{\prime}-15 y=x+e^{2 x}$

Use a table of Laplace Transforms to find the Laplace transform of each function $\mathrm{f}(\mathrm{t})$ in the problems below. - $f(t)=\frac{t^{4}}{24}+1-\cos 6 t$

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 $y^{\prime}(0)=26$

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.

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 $y^{\prime}(0)=0$

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

For the problems below, solve each differential equation. - $y^{\prime \prime}-4 y^{\prime}+29 y=0$

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: $\frac{d^{2} i}{d t^{2}}+\frac{R d i}{L d t}+\frac{1}{C L} i=0$ )

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

For the problems below, solve each differential equation. - $\mathrm{y}^{\prime \prime}+\mathrm{y}=4 \sin \mathrm{x}$

For the problems below, solve each differential equation. - $y^{\prime \prime}+8 y^{\prime}=0$

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

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

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

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 $\mathrm{x}=0$ .

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

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

Use a table of Laplace Transforms to find the Laplace transform of each function $\mathrm{f}(\mathrm{t})$ in the problems below. - $f(t)=e^{4 t} \cos 8 t+5 t e^{10 t}$

Analytic Geometry

The Derivative

Applications of the Derivative

Derivatives of Transcendental Functions

The Integral

Applications of Integration

Methods of Integration

Three-Space: Partial Derivatives and Double Integrals

Progressions and the Binomial Theorem

Series

First-Order Differential Equations

Filters

Exam 12: Second-Order Differential Equations

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

For the problems below, solve each differential equation. - y′′+2y′−15y=x+e2xy^{\prime \prime}+2 y^{\prime}-15 y=x+e^{2 x}y′′+2y′−15y=x+e2x

Use a table of Laplace Transforms to find the Laplace transform of each function f(t)\mathrm{f}(\mathrm{t})f(t) in the problems below. - f(t)=t424+1−cos⁡6tf(t)=\frac{t^{4}}{24}+1-\cos 6 tf(t)=24t4​+1−cos6t

For the problems below, solve each differential equation subject to the given conditions by using Laplace transforms. - y′′−5y′−24y=0;y(0)=17y^{\prime \prime}-5 y^{\prime}-24 y=0 ; y(0)=17y′′−5y′−24y=0;y(0)=17 and y′(0)=26y^{\prime}(0)=26y′(0)=26

A spring is stretched 6in6 \mathrm{in}6in . by a weight of 10lb10 \mathrm{lb}10lb . If the weight is displaced a distance of 4in4 \mathrm{in}4in . from a rest position and then released, find the equation of motion.

For the problems below, solve each differential equation subject to the given conditions by using Laplace transforms. - y′′−8y′+16y=te4t;y(0)=0y^{\prime \prime}-8 y^{\prime}+16 y=t e^{4 t} ; y(0)=0y′′−8y′+16y=te4t;y(0)=0 and y′(0)=0y^{\prime}(0)=0y′(0)=0

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

For the problems below, solve each differential equation. - y′′−4y′+29y=0y^{\prime \prime}-4 y^{\prime}+29 y=0y′′−4y′+29y=0

Find the particular solution subject to the given conditions. y′′+81y=0;y=7 and y′=−27 when x=π18\mathrm{y}^{\prime \prime}+81 \mathrm{y}=0 ; \mathrm{y}=7 \text { and } \mathrm{y}^{\prime}=-27 \text { when } \mathrm{x}=\frac{\pi}{18}y′′+81y=0;y=7 and y′=−27 when x=18π​

For the problems below, solve each differential equation. - y′′+y=4sin⁡x\mathrm{y}^{\prime \prime}+\mathrm{y}=4 \sin \mathrm{x}y′′+y=4sinx

For the problems below, solve each differential equation. - y′′+8y′=0y^{\prime \prime}+8 y^{\prime}=0y′′+8y′=0

Using a table of Laplace transforms, find the inverse transform of F(s)=1s2+10s+29F(s)=\frac{1}{s^{2}+10 s+29}F(s)=s2+10s+291​ .

Find the general solution of the differential equation y′′−4y=8e2xy^{\prime \prime}-4 y=8 e^{2 x}y′′−4y=8e2x .

Find the inverse transform for the function F(s)F(s)F(s) . F(s)=9s2+81+1(s+5)3F(s)=\frac{9}{s^{2}+81}+\frac{1}{(s+5)^{3}}F(s)=s2+819​+(s+5)31​

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

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

Find the general solution of the differential equation y′′+4y′+3y=20cos⁡xy^{\prime \prime}+4 y^{\prime}+3 y=20 \cos xy′′+4y′+3y=20cosx .

Use a table of Laplace Transforms to find the Laplace transform of each function f(t)\mathrm{f}(\mathrm{t})f(t) in the problems below. - f(t)=e4tcos⁡8t+5te10tf(t)=e^{4 t} \cos 8 t+5 t e^{10 t}f(t)=e4tcos8t+5te10t

Analytic Geometry

The Derivative

Applications of the Derivative

Derivatives of Transcendental Functions

The Integral

Applications of Integration

Methods of Integration

Three-Space: Partial Derivatives and Double Integrals

Progressions and the Binomial Theorem

Series

First-Order Differential Equations

Filters

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

For the problems below, solve each differential equation. - $y^{\prime \prime}+2 y^{\prime}-15 y=x+e^{2 x}$

Use a table of Laplace Transforms to find the Laplace transform of each function $\mathrm{f}(\mathrm{t})$ in the problems below. - $f(t)=\frac{t^{4}}{24}+1-\cos 6 t$

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 $y^{\prime}(0)=26$

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.

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 $y^{\prime}(0)=0$

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

For the problems below, solve each differential equation. - $y^{\prime \prime}-4 y^{\prime}+29 y=0$

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

For the problems below, solve each differential equation. - $\mathrm{y}^{\prime \prime}+\mathrm{y}=4 \sin \mathrm{x}$

For the problems below, solve each differential equation. - $y^{\prime \prime}+8 y^{\prime}=0$

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

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

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

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 $\mathrm{x}=0$ .

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

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

Use a table of Laplace Transforms to find the Laplace transform of each function $\mathrm{f}(\mathrm{t})$ in the problems below. - $f(t)=e^{4 t} \cos 8 t+5 t e^{10 t}$