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

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

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

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

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

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.

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

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

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

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.

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

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

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.

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

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

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.

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

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

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

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.

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