Deck 36: Solving Differential Equations by the Laplace Transform and by Numerical Methods

Question 1

Find the Laplace transform of $f(t)=\sin 4 t$ by direct integration.

Answer 1

$\frac{4}{s^{2}+16}$

Question 2

Use a table to find the Laplace transform of $f(t)=3 e^{2 t}-e^{-t}$ .

Answer 2

$\frac{3}{s-2}-\frac{1}{s+1}$

Question 3

Use a table to find the Laplace transform of $f(t)=t e^{2 t}-t \cos 4 t$ .

Answer 3

$\frac{1}{(s-2)^{2}}-\frac{s^{2}-16}{\left(s^{2}+16\right)^{2}}$

Question 4

Find the Laplace transform of $y^{\prime \prime}-4 y^{\prime}+4 y$ , and substitute in the initial conditions, $y(0)=6, y^{\prime}(0)=0$ .

Question 5

Find the Laplace transform of the function: $f(t)=\sin 5 t+2 t e^{t}$

Question 6

Find the Laplace transform of the expression and substitute in the initial conditions: $2 y^{\prime \prime}+3 y^{\prime}+y$ ; $y^{\prime}(0,1)=2$

Question 7

Find the Laplace transform of the function $f(t)=2 t-6$ .

Question 8

Find the Laplace transform of the function $f(t)=t e^{3 t}+2 e^{7 t}$ .

Question 9

Find the Laplace transform of $2 y^{\prime \prime}+5 y^{\prime}-3 y$ and substitute the initial condition $y^{\prime}(0,0)=0$ .

Question 10

Find the Laplace transform of $y^{\prime \prime}+2 y^{\prime}+3 y$ and substitute the initial condition $y^{\prime}(0,6)=7$ .

Question 11

Find the Laplace transform of $f(t)=5 t^{3}$ by direct integration.

Question 12

Find the Laplace transform of $4 y^{\prime}-7 y$ , and substitute in the initial condition $y(0)=5$ .

Question 13

Use a table to find the Laplace transform of $f(t)=2 e^{-4 t} \cos 3 t$ .

Question 14

Find the Laplace transform of $f(t)=2 t^{2} e^{4 t}$ .

Question 15

Find the Laplace transform of $2 y^{\prime \prime}-5 y^{\prime}-7 y$ , and substitute in the initial condition $y^{\prime}(0,4)=3$ .

Question 16

Find the Laplace transform of $f(t)=\sin 2 t+2 t \cos 2 t$ .

Question 17

Find the Laplace transform of $4 y^{\prime \prime}-3 y$ , and substitute in the initial condition $y^{\prime}(0,0)=5$ .

Question 18

Find the inverse transform of $\frac{5}{S}$ .

Question 19

Find the inverse transform of $\frac{1}{s^{2}+4 s+13}$ .

Question 20

Find the inverse transform of $\frac{10s-3}{s(s+2)(s-3)}$ .

Question 21

Find the inverse of the transform: $\frac{s+1}{s^{3}+s}$

Question 22

Find the inverse Laplace transform of the function $\frac{2}{s-2 s-15}$ .

Question 23

Find the inverse Laplace Transform of the function $\frac{16-3 s^{2}}{s^{3}+16 s}$ .

Question 24

Find the inverse transform of $\frac{s}{s^{2}+3 s+2}$ .

Question 25

Find the inverse transform of $\frac{4 s}{s^{2}+9}$ .

Question 26

Find the inverse transform of $\frac{s}{\left(s^{2}+16\right)^{2}}$ .

Question 27

Find the inverse transform of $\frac{s+4}{s^{2}+8 s+20}$ .

Question 28

Solve the following differential equation by the Laplace transform: $y^{\prime}+6 y=12 t, y(0)=0$

Question 29

Solve the following differential equation by the Laplace transform: $y^{\prime \prime}-4 y^{\prime}=6, y(0)=1, y^{\prime}(0)=2$

Question 30

Solve the following differential equation by the Laplace transform: $5 y^{\prime \prime}-4 y^{\prime}=\cos 2 t, y(0)=2, y^{\prime}(0)=2$

Question 31

Solve the differential equation by the Laplace transform: $y^{\prime \prime}-4 y^{\prime}-5 y=4 t ; y^{\prime}(0,2)=1$

Question 32

Solve the differential equation by the Laplace transform: $2 y^{\prime}+4 y=\operatorname{cost} ; y(0)=0$

Question 33

Solve the differential equation by the Laplace transform: $2 y^{\prime \prime}-5 y^{\prime}=e^{t} ; y(0)=0, y^{\prime}(0)=0$

Question 34

Solve the differential equation by the Laplace transform: $y^{\prime \prime}-4 y=e^{2 t} ; y^{\prime}(0,0)=5$

Question 35

Solve the differential equation by the Laplace transform: $4 y^{\prime}+y=\sin t ; y(0)=0$

Question 36

Solve the differential equation by the Laplace transform: $y^{\prime \prime}+y^{\prime}=2 ; y^{\prime}(0,1)=5$

Question 37

Solve the differential equation by the Laplace transform: $y^{\prime}-8 y=1-e^{-t} ; y(0)=2$

Question 38

In an RL circuit, $R=1750 \Omega$ and $L=0.290 \mathrm{H}$ . The circuit is connected to a DC source of $115 \mathrm{~V}$ at $t=0$ . If $i$ is zero at $t=0$ , find the current.

Question 39

In an RL circuit, $R=25 \Omega$ and $L=0.25 \mathrm{H}$ . The circuit is connected to an $\mathrm{AC}$ source of $50 \sin 100 \mathrm{t} \mathrm{V}$ at $t=0$ . If $i$ is zero at $t=0$ , find the current.

Question 40

The current in an RLC circuit satisfies the equation $0.75 i^{\prime \prime}+450 i^{\prime}+9000 i=150$ . If $i$ and $i^{\prime}$ are zero at $t=0$ , find the equation for the current.

Question 41

In an RL circuit, $R=250 \Omega$ and $L=0.500 \mathrm{H}$ . The circuit is connected to a DC source with an initial voltage of $75 \mathrm{~V}$ . If the initial current is $0 \mathrm{~A}$ , find the current.

Question 42

Use any numerical method to solve the following differential equation: $e^{\mathrm{x}} \mathrm{y}^{\prime}-x^{2} y-y=4 ; y(1)=7$ . Find $y(2)$ to two decimal places.

Question 43

Use any numerical method to solve the following differential equation: $y^{\prime}-\cos (x y)=\ln (x y) ; y(3)=7$ . Find $y(4)$ to two decimal places.

Question 44

Use any numerical method to solve the differential equation: $y^{\prime}+x \sin y=x ; y(1)=2$ . Find $y(2)$ to two decimal places.

Question 45

Use any numerical method to solve the differential equation: $y^{\prime}=e^{x y}-5 ; y(0)=2$ . Find $y(1)$ to three decimal places.

Question 46

Use any numerical method to solve the differential equation: $y^{\prime}+x \sin y=x ; y(0)=5$ . Find $y(1)$ to three decimal places.

Question 47

Solve the differential equation $y^{\prime}=y^{3}+x, y(0)=0$ with a step size of 0.1 , using Euler's method. Find $y(1)$ to three decimal places.

Question 48

Solve the differential equation $y^{\prime}=1+y$ , with initial condition $y(0)=1$ and step size 0.1 , using the modified Euler's method. Find $y(1)$ three decimal places.

Question 49

Use any numerical method to solve the differential equation: $2 y^{\prime}+4 y^{2}=5 x$ . If $y(0)=2$ , find $y(1)$ to three decimal places.

Question 50

Solve the differential equation $y^{\prime}+y=x+1$ , with initial condition $y(1)=2$ and step size 0.1 , using the modified Euler's method. Find $y(2)$ to three decimal places.

Question 51

Solve the differential equation $x y^{\prime}-y=1$ , with initial condition $y(2)=4$ and step size 0.1 , using the modified Euler's method. Find $y(3)$ to three decimal places.

Question 52

Use any numerical method to solve the following differential equation: $y^{\prime \prime}+x^{2}+y y^{\prime}=10 ; y^{\prime}(1,7)=3$ . Find $y(2)$ to three decimal places.

Question 53

Use any numerical method to solve the following differential equation: $\frac{y^{\prime \prime}}{x^{2}}+y^{\prime} \sin x=0 ; y^{\prime}(1,1)=1$ . Find $y(2)$ to three decimal places.

Question 54

Use any numerical method to solve the differential equation: $y^{\prime \prime}+x y^{\prime}+x^{2} y=0 ; y^{\prime}=2$ at $(1,1)$ . Find $y(2)$ to three decimal places.

Question 55

Solve by the Runge-Kutta method. Find the value of $y$ at $x=2$ . Take a step size of 0.1 unit.
$y^{\prime \prime}+y^{\prime}+2 x y=8, y^{\prime}(1,2)=4$

Question 56

Solve by the Runge-Kutta method. Find the value of $y(3)$ . Take a step size of 0.1 unit.
$y^{\prime \prime}+y^{\prime}+4 x y=5, y^{\prime}(2,0)=1$