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

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

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

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

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.

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

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.

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

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

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

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

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

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.

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

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

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

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

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

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.

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

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