Chapter 7: The Laplace transformContent check — Differential Equations (Lebl), Chapter 7: The Laplace transform

Find the Laplace transform of $4{(t+1)}^2$⁠.

Find the inverse Laplace transform of $\frac{8}{s^3(s+2)}$⁠.

Find the Laplace transform of $t{e}^{-t}$⁠. Hint: Shifting property.

Find the Laplace transform of $\sin (t)e^{-t}$⁠. Hint: Integrate by parts.

Using the Heaviside function $u(t)$⁠, write down the function

$$f(t)=\{\begin{array}{cc}0\hfill & \text{if }\phantom{1\le }t<1,\hfill \\ t-1\hfill & \text{if }1\le t<2,\hfill \\ 1\hfill & \text{if }2\le t.\hfill \end{array}$$

Solve $x^{\prime \prime }-x=(t^2-1)u(t-1)$ for initial conditions $x(0)=1$⁠, $x^{\prime }(0)=2$ using the Laplace transform.

Find the transfer function for $x^{\prime }+x=f(t)$⁠, $x(0)=0$⁠.

Suppose $Lx=f(t)$⁠, $x(0)=0$⁠, $x^{\prime }(0)=0$ for the input function $f(t)=4$ gives the output $x(t)=1-\cos (2t)$⁠. Find $F(s)$⁠, $X(s)$⁠, and the transfer function $H(s)$⁠.

Let $f(t)=\cos t$ for $t\ge 0$⁠, and $g(t)=e^{-t}$⁠. Compute $f\ast g$⁠.

Compute ${ℒ}^{-1}\left\{\frac{5}{s^4+s^2}\right\}$ using convolution.

Solve $x^{\prime \prime }+x=\sin t$⁠, $x(0)=0$⁠, $x^{\prime }(0)=0$ using convolution.

Solve $x^{\prime \prime \prime }+x^{\prime }=f(t)$⁠, $x(0)=0$⁠, $x^{\prime }(0)=0$⁠, $x^{\prime \prime }(0)=0$ using convolution. Write the result as a definite integral.

Solve (find the impulse response) $x^{\prime \prime }=\delta (t)$⁠, $x(0)=0$⁠, $x^{\prime }(0)=0$⁠.

Solve (find the impulse response) $x^{\prime }+ax=\delta (t)$⁠, $x(0)=0$⁠.

Suppose that $Lx=\delta (t)$⁠, $x(0)=0$⁠, $x^{\prime }(0)=0$⁠, has the solution $x(t)=e^t\sin (t)$ for $t>0$⁠. Find (in closed form) the solution to $Lx=e^t$⁠, $x(0)=0$⁠, $x^{\prime }(0)=0$ for $t>0$⁠.

Compute ${ℒ}^{-1}\left\{\frac{s^2}{s^2+1}\right\}$⁠.

Compute ${ℒ}^{-1}\left\{\frac{3s^2{e}^{-s}+2}{s^2}\right\}$⁠.

Solve

$$\begin{array}{cc}\hfill & y_t+y_x=1,0<x<\infty ,t>0,\hfill \\ \hfill & y(0,t)=0,y(x,0)=0.\hfill \end{array}$$

For a $c>0$⁠, solve

$$\begin{array}{cc}\hfill & y_t+y_x+cy=0,0<x<\infty ,t>0,\hfill \\ \hfill & y(0,t)=\sin (t),y(x,0)=0.\hfill \end{array}$$

Find the corresponding ODE problem for $Y(x)$⁠, after transforming the $t$ variable

$$\begin{array}{cc}\hfill & y_{tt}+3y_{xx}+y=x+t,-1<x<1,t>0,\hfill \\ \hfill & y(-1,t)=0,y(1,t)=0,y(x,0)=(1-x^2),y_t(x,0)=0.\hfill \end{array}$$

Do not solve the problem.

Use the Laplace transform in $t$ to solve

$$\begin{array}{cc}\hfill & y_{tt}=y_{xx},-\infty <x<\infty ,t>0,\hfill \\ \hfill & y(x,0)=\sin (x),y_t(x,0)=0.\hfill \end{array}$$

Assume that $y$ is bounded. Hint: Note that for any fixed $s>0$⁠, $e^{sx}$ blows up as $x\to +\infty$ and $e^{-sx}$ blows up as $x\to -\infty$⁠.

Use the Laplace transform in $t$ to solve

$$\begin{array}{cc}\hfill & y_t=y_{xx},0<x<\infty ,t>0,\hfill \\ \hfill & y(0,t)=f(t),y(x,0)=0,\hfill \end{array}$$

where $f(t)$ is some function. Assume that $y$ is bounded. Give the answer as a convolution.