5.2E: Exercises for Section 5.2

Exercise \(\PageIndex{1}\)

Verify the Laplace transforms given in Table 5.2.1 by deriving them using the definition of the Laplace transform.

Exercise \(\PageIndex{2}\)

Using the Heaviside function \[u(t)= \left\{ \begin{array}{cc} 0 & {\rm{if~}}t<0, \\ 1 & {\rm{if~}}t \geq 0.\end{array} \right. \nonumber \] write down the piecewise function that is \(0\) for \(t<0, t^2\) for \(t\) in \([0,1]\) and \(t\) for \(t>1\).

Answer

\(f(t) = t^2 u(t) + t(1-t)u(t-1)\)

Exercise \(\PageIndex{3}\)

Using the Laplace transform solve

\[ mx'' + cx'+kx =0,\quad x(0)=a, \quad x'(0)=b.\nonumber\]

where \(m>0,c>0,k>0\), and \(c^2-4km>0\) (system is overdamped).

Exercise \(\PageIndex{4}\)

Using the Laplace transform solve

\[ mx'' + cx'+kx =0,\quad x(0)=a, \quad x'(0)=b.\nonumber\]

where \(m>0,c>0,k>0\), and \(c^2-4km<0\) (system is underdamped).

Answer

\(x(t) = e^{-\frac{c}{2m}t} \left( a \cos\left(\frac{\sqrt(4km-c^2}{2m}\right) + \frac{2m b + c a}{\sqrt(4km-c^2)} \sin\left(\frac{\sqrt(4km-c^2}{2m}\right)t\right). \)

Exercise \(\PageIndex{5}\)

Using the Laplace transform solve

\[ mx'' + cx'+kx =0,\quad x(0)=a, \quad x'(0)=b.\nonumber\]

where \(m>0,c>0,k>0\), and \(c^2=4km\) (system is critically damped).

Exercise \(\PageIndex{6}\)

Solve \(x''+x=u(t-1)\) for initial conditions \(x(0)=0\) and \(x'(0)=0\).

Answer

\(x(t) = u(t - 1) (1 - \cos(t - 1))\)

Exercise \(\PageIndex{7}\)

Suppose \(\mathcal{L}\{f(t)\}=F(s)\), then show

\[ \mathcal{L}\{-tf(t)\}=F'(s).\nonumber\]

Hint: Differentiate under the integral sign.

Exercise \(\PageIndex{8}\)

Solve \(x'''+x=t^3u(t-1)\) for initial conditions \(x(0)=1\) and \(x'(0)=0\), \(x''(0)=0\).

Answer

\(x(t) = \frac{2}{3}e^{\frac{1}{2}t}\cos(\frac{\sqrt3}{2}t)+\frac{1}{3}e^{-t}+4(t-1)[-5+3(t-1)+3(t-1)^2+(t-1)^3+\frac{2}{3}e^{-(t-1)}+\frac{13}{2}e^{\frac{1}{2}(t-1)}\cos(\frac{\sqrt 3(t-1)}{2} -3\sqrt 3e^{\frac{1}{2}(t-1)}\sin(\frac{\sqrt 3(t-1)}{2}]\)

Exercise \(\PageIndex{9}\)

Show the second shifting property: \( \mathcal{L}\{f(t-a)u(t-a)\}=e^{-as}\mathcal{L}\{f(t)\} \).

Exercise \(\PageIndex{10}\)

Define

\[ f(t)= \left\{ \begin{array}{ccc} (t-1)^2 & if~1 \leq t<2, \\ 3-t & if~2 \leq t<3, \\ 0 & otherwise. \end{array} \right. \nonumber\]

1. Sketch the graph of \(f(t)\).
2. Write down \(f(t)\) using the Heaviside function.
3. Solve \(x''+x=f(t), x(0)=0,x'(0)=0\) using Laplace transform.

Answer

1. 

2. \(f(t)= (t-1)^2[u(t-1)-u(t-2)]+(3-t)[u(t-2)-u(t-3)]\)
3. \(x(t) =u_1(t)[-2+(t-1)^2+2\cos(t-1)]+u_2(t)[2-3(t-2)-9t-2)^2-2\cos(t-2)+3\sin(t-2)]+u_3(t)[t-3-\sin(t-3)]\)

Exercise \(\PageIndex{11}\)

Find the transfer function for \(mx'' + cx'+kx =f(t)\) (assuming the initial conditions are zero).

Exercise \(\PageIndex{12}\)

Solve \(x''-x=(t^2-1)u(t-1)\) for initial conditions \(x(0)=1,x'(0)=2\) using the Laplace transform.

Answer

\(x(t)=(2e^{t-1}-t^{2}-1)u(t-1)-\frac{1}{2}e^{-t}+\frac{3}{2}e^{t}\)

Exercise \(\PageIndex{13}\)

Find the transfer function for \(x'+x=f(t)\) (assuming the initial conditions are zero).

Answer

\(H(s)=\frac{1}{s+1}\)