5.3E: Exercises for Section 5.3

Exercise \(\PageIndex{1}\)

Let \(f(t)=t^2\) for \(t \geq 0\), and \(g(t)=u(t-1)\). Compute \(f * g\).

Exercise \(\PageIndex{2}\)

Let \(f(t)=t\) for \(t \geq 0\), and \(g(t)=\sin t\) for \(t \geq 0\). Compute \(f * g\).

Answer

\((f * g)(t)= sint - t\) for \(t \ge 0\)

Exercise \(\PageIndex{3}\)

Find the solution to

\( mx''+cx'+kx=f(t),\quad x(0)=0,\quad x'(0)=0,\)

for an arbitrary function \(f(t)\), where \(m>0,c>0,k>0\), and \(c^2-4km>0\) (system is overdamped). Write the solution as a definite integral.

Exercise \(\PageIndex{4}\)

Find the solution to

\( mx''+cx'+kx=f(t),\quad x(0)=0,\quad x'(0)=0,\)

for an arbitrary function \(f(t)\), where \(m>0,c>0,k>0\), and \(c^2-4km<0\) (system is underdamped). Write the solution as a definite integral.

Answer

\( x(t) = \frac {2}{\sqrt(4km-c^2)}\int_0^t e^{-\frac {c}{2m}t} \sin (\frac{\sqrt (4km-c^2)}{2m}t)f(t-\tau) d\tau \)

Exercise \(\PageIndex{5}\)

Find the solution to

\( mx''+cx'+kx=f(t),\quad x(0)=0,\quad x'(0)=0,\)

for an arbitrary function \(f(t)\), where \(m>0,c>0,k>0\), and \(c^2=4km\) (system is critically damped). Write the solution as a definite integral.

Exercise \(\PageIndex{6}\)

Solve

\( x(t)=e^{-t} +\int_0^t\cos(t-\tau)x(\tau)~d\tau . \)

Answer

\( x(t) = e^{-t}+\sin t +\cos t -1 \)

Exercise \(\PageIndex{7}\)

Solve

\( x(t)=\cos t +\int_0^t\cos(t-\tau)x(\tau)~d\tau . \)

Answer

\( x(t) = \sin t + 2\cos t -1 \)

Exercise \(\PageIndex{8}\)

Compute \(\mathcal{L}^{-1} \left\{ \frac{s}{(s^2+4)^2}\right\}\) using convolution.

Answer

\( \mathcal{L}^{-1} \left\{ \frac{s}{(s^2 + 4)^2} \right\} = \frac{1}{4} t \sin(2t). \)

Exercise \(\PageIndex{9}\)

Write down the solution to \(x''-2x=e^{-t^2},x(0)=0,x'(0)=0\) as a definite integral. Hint: Do not try to compute the Laplace transform of \(e^{-t^2}\).

Exercise \(\PageIndex{10}\)

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

Answer

\(\frac{1}{2}(\cos t+\sin t-e^{-t})\)

Exercise \(\PageIndex{11}\)

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

Answer

\(5t-5\sin t\)

Exercise \(\PageIndex{12}\)

Solve \(x''+x=\sin t, x(0)=0, x'(0)=0\) using convolution.

Answer

\(\frac{1}{2}(\sin t-t\cos t)\)

Exercise \(\PageIndex{13}\)

Solve \(x'''+x'=f(t), x(0)=0, x'(0)=0,x''(0)=0\) using convolution. Write the result as a definite integral.

Answer

\(\int_{0}^{t} f(\tau )(1-\cos (t-\tau ))d\tau \)