5.7: Problems

Problems

1. Find the Laplace transform of the following functions:

1. \(f(t)=9 t^{2}-7\)
2. \(f(t)=e^{5 t-3}\).
3. \(f(t)=\cos 7 t\).
4. \(f(t)=e^{4 t} \sin 2 t\).
5. \(f(t)=e^{2 t}(t+\cosh t)\).
6. \(f(t)=t^{2} H(t-1)\).
7. \(f(t)=\left\{\begin{array}{cl}\sin t, & t<4 \pi, \\ \sin t+\cos t, & t>4 \pi .\end{array}\right.\)
8. \(f(t)=\int_{0}^{t}(t-u)^{2} \sin u d u\).
9. \(f(t)=\int_{0}^{t} \cosh u d u\).
10. \(f(t)=(t+5)^{2}+t e^{2 t} \cos 3 t\) and write the answer in the simplest form.

2. Find the inverse Laplace transform of the following functions using the properties of Laplace transforms and the table of Laplace transform pairs.

1. \(F(s)=\dfrac{18}{s^{3}}+\dfrac{7}{s}\)
2. \(F(s)=\dfrac{1}{s-5}-\dfrac{2}{s^{2}+4}\).
3. \(F(s)=\dfrac{s+1}{s^{2}+1}\).
4. \(F(s)=\dfrac{3}{s^{2}+2 s+2}\).
5. \(F(s)=\dfrac{1}{(s-1)^{2}}\).
6. \(F(s)=\dfrac{e^{-3 s}}{s^{2}-1}\)
7. \(F(s)=\dfrac{1}{s^{2}+4 s-5}\)
8. \(F(s)=\dfrac{s+3}{s^{2}+8 s+17}\).

3. Use Laplace transforms to solve the following initial value problems. Where possible, describe the solution behavior in terms of oscillation and decay.
   1. \(y^{\prime \prime}-5 y^{\prime}+6 y=0, y(0)=2, y^{\prime}(0)=0\)
   2. \(y^{\prime \prime}+2 y^{\prime}+5 y=0, y(0)=1, y^{\prime}(0)=0\).
   3. \(y^{\prime \prime}-y=t e^{2 t}, y(0)=0, y^{\prime}(0)=1\).
   4. \(y^{\prime \prime}-3 y^{\prime}-4 y=t^{2}, y(0)=2, y^{\prime}(0)=1\).
   5. \(y^{\prime \prime \prime}-3 y^{\prime}-2 y=e^{t}, y(0)=1, y^{\prime}(0)=0\).
4. Use Laplace transforms to solve the following initial value problems. Where possible, describe the solution behavior in terms of oscillation and decay.
   1. \(y^{\prime \prime}+4 y=\delta(t-1), y(0)=3, y^{\prime}(0)=0\)

   2. \(y^{\prime \prime}-4 y^{\prime}+13 y=\delta(t-1), y(0)=0, y^{\prime}(0)=2\).
      

   3. \(y^{\prime \prime}+6 y^{\prime}+18 y=2 H(\pi-t), y(0)=0, y^{\prime}(0)=0\).
   4. \(y^{\prime \prime}+4 y=f(t), y(0)=1, y^{\prime}(0)=0\), where \(f(t)=\left\{\begin{array}{cc}1, & 0<t<1 \\ 0, & t>1\end{array}\right.\)
5. For the following problems, draw the given function and find the Laplace transform in closed form.
   1. \(f(t)=1+\sum_{n=1}^{\infty}(-1)^{n} H(t-n)\).
   2. \(f(t)=\sum_{n=0}^{\infty}[H(t-2 n+1)-H(t-2 n)]\).
   3. \[\begin{aligned} f(t)=& \sum_{n=0}^{\infty}(t-2 n)[H(t-2 n)-H(t-2 n-1)] \\ &+\sum_{n=0}^{\infty}(2 n+2-t)[H(t-2 n-1)-H(t-2 n-2)] \end{aligned} \nonumber \]

6. The period, \(T\), and the function defined on its first period are given. Sketch several periods of these periodic functions. Make use of the periodicity to find the Laplace transform of each function.
   1. \(f(t)=\sin t, T=2 \pi\).
   2. \(f(t)=t, T=1\).
   3. \(f(t)=\left\{\begin{array}{cl}t, & 0 \leq t \leq 1, \\ 2-t, & 1 \leq t \leq 2,\end{array} T=2\right.\).
   4. \(f(t)=t[H(t)-H(t-1)], T=2\).
   5. \(f(t)=\sin t[H(t)-H(t-\pi)], T=\pi\).
7. Compute the convolution \((f * g)(t)\) (in the Laplace transform sense) and its corresponding Laplace transform \(\mathcal{L}[f * g]\) for the following functions:
   1. \(f(t)=t^{2}, g(t)=t^{3}\)
   2. \(f(t)=t^{2}, g(t)=\cos 2 t\).
   3. \(f(t)=3 t^{2}-2 t+1, g(t)=e^{-3 t}\).
   4. \(f(t)=\delta\left(t-\dfrac{\pi}{4}\right), g(t)=\sin 5 t\).
8. Use the Convolution Theorem to compute the inverse transform of the following:
   1. \(F(s)=\dfrac{2}{s^{2}\left(s^{2}+1\right)}\).
   2. \(F(s)=\dfrac{e^{-3 s}}{s^{2}}\).
   3. \(F(s)=\dfrac{1}{s\left(s^{2}+2 s+5\right)}\).
9. Find the inverse Laplace transform in two different ways: (i) Use tables. (ii) Use the Convolution Theorem.
   1. \(F(s)=\dfrac{1}{s^{3}(s+4)^{2}}\).
   2. \(F(s)=\dfrac{1}{s^{2}-4 s-5}\).
   3. \(F(s)=\dfrac{s+3}{s^{2}+8 s+17}\).
   4. \(F(s)=\dfrac{s+1}{(s-2)^{2}(s+4)}\).
   5. \(F(s)=\dfrac{s^{2}+8 s-3}{\left(s^{2}+2 s+1\right)\left(s^{2}+1\right)}\).
10. A linear Volterra integral equation, introduced by Vito Volterra ( \(1860-\) \(1940)\), is of the form

\[y(t)=f(t)+\int_{0}^{t} K(t-\tau) y(\tau) d \tau\nonumber \]

where \(y(t)\) is an unknown function and \(f(t)\) and the "kernel," \(K(t)\), are given functions. The integral is in the form of a convolution integral and such equations can be solved using Laplace transforms. Solve the following Volterra integral equations.

1. \(y(t)=e^{-t}+\int_{0}^{t} \cos (t-\tau) y(\tau) d \tau\)
2. \(y(t)=t-\int_{0}^{t}(t-\tau) y(\tau) d \tau\).
3. \(y(t)=t+2 \int_{0}^{t} e^{t-\tau} y(\tau) d \tau\).
4. \(\sin t=\int_{0}^{t} e^{t-\tau} y(\tau) d \tau\). Note: This is a Volterra integral equation of the first kind.

11. Use Laplace transforms to convert the following system of differential equations into an algebraic system and find the solution of the differential equations.

\[\begin{array}{llll} x^{\prime \prime} & =3 x-6 y, & x(0) & =1, & x^{\prime}(0) & =0 \\ y^{\prime \prime} & =x+y, & y & (0)=0, & y^{\prime}(0) & =0 \end{array} \nonumber \]

12. Use Laplace transforms to convert the following nonhomogeneous systems of differential equations into an algebraic system and find the solutions of the differential equations.
    1. \[\begin{aligned} &x^{\prime}=2 x+3 y+2 \sin 2 t, \quad x(0)=1 \\ &y^{\prime}=-3 x+2 y, \quad y(0)=0 \end{aligned} \nonumber \]

    2. \[\begin{aligned} &x^{\prime}=-4 x-y+e^{-t}, \quad x(0)=2 \\ &y^{\prime}=x-2 y+2 e^{-3 t}, \quad y(0)=-1 \end{aligned} \nonumber \]

    3. \[\begin{aligned} &x^{\prime}=x-y+2 \cos t, \quad x(0)=3 \\ &y^{\prime}=x+y-3 \sin t, \quad y(0)=2 \end{aligned} \nonumber \]

13. Redo Example \(5.19\) using the values \(R_{1}=1.00 \Omega, R_{2}=1.40 \Omega, L_{1}=0.80\) \(\mathrm{H}, L_{2}=1.00 \mathrm{H}\). and \(v_{0}=100 \mathrm{~V}\) in \(v(t)=v_{0}\left(1-H\left(t-t_{0}\right)\right)\). Plot the currents as a function of time for several values of \(t_{0}\).