8.E: The Laplace Transform (Exercises)
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These are homework exercises to accompany Libl's " Differential Equations for Engineering " Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Prerequisite for the course is the basic calculus sequence.
6.1: The Laplace transform
Exercise \(\PageIndex{6.1.1}\)
Find the Laplace transform of \(3+ t^5 + \sin (\pi t)\).
Exercise \(\PageIndex{6.1.2}\)
Find the Laplace transform of \(a + bt +ct^2\) for some constants \(a\) , \(b\) , and \(c\) .
Exercise \(\PageIndex{6.1.3}\)
Find the Laplace transform of \(A \cos (\omega t) + B \sin (\omega t ) \).
Exercise \(\PageIndex{6.1.4}\)
Find the Laplace transform of \( \cos^2 (\omega t ) \).
Exercise \(\PageIndex{6.1.5}\)
Find the inverse Laplace transform of \(\dfrac{4}{s^2-9}\).
Exercise \(\PageIndex{6.1.6}\)
Find the inverse Laplace transform of \( \dfrac{2s}{s^2-1}\).
Exercise \(\PageIndex{6.1.7}\)
Find the inverse Laplace transform of \( \dfrac{1}{(s-1)^2(s+1)}\).
Exercise \(\PageIndex{6.1.8}\)
Find the Laplace transform of \(f(t)= \left\{ \begin{array}{cc} t & {\rm{if~}}t \geq 1, \\ 0 & {\rm{if~}}t < 1.\end{array} \right.\).
Exercise \(\PageIndex{6.1.9}\)
Find the inverse Laplace transform of \( \dfrac{s}{(s^2+s+2)(s+4)}\).
Exercise \(\PageIndex{6.1.10}\)
Find the Laplace transform of \(\sin \left( \omega (t-a) \right) \).
Exercise \(\PageIndex{6.1.11}\)
Find the Laplace transform of \( t \sin (\omega t) \) . Hint: Several integrations by parts.
Exercise \(\PageIndex{6.1.12}\)
Find the Laplace transform of \(4(t+1)^2\).
- Answer
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\(\frac{8}{s^{3}}+\frac{8}{s^{2}}+\frac{4}{s}\)
Exercise \(\PageIndex{6.1.13}\)
Find the inverse Laplace transform of \(\dfrac{8}{s^3 (s+2)}\).
- Answer
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\(2t^{2}-2t+1-e^{-2t}\)
Exercise \(\PageIndex{6.1.14}\)
Find the Laplace transform of \(te^{-t}\) (Hint: integrate by parts).
- Answer
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\(\frac{1}{(s+1)^{2}}\)
Exercise \(\PageIndex{6.1.15}\)
Find the Laplace transform of \(\sin (t) e^{-t}\) (Hint: integrate by parts).
- Answer
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\(\frac{1}{s^{2}+2s+2}\)
6.2: Transforms of Derivatives and ODEs
Exercise \(\PageIndex{6.2.1}\)
Verify Table 6.2.1 .
Exercise \(\PageIndex{6.2.2}\)
Using the Heaviside function write down the piecewise function that is \(0\) for \(t<0, t^2\) for \(t\) in \([0,1]\) and \(t\) for \(t>1\).
Exercise \(\PageIndex{6.2.3}\)
Using the Laplace transform solve
\[ mx'' + cx'+kx =0,\quad x(0)=a, \quad x'(0)=b.\]
where \(m>0,c>0,k>0\), and \(c^2-4km>0\) (system is overdamped).
Exercise \(\PageIndex{6.2.4}\)
Using the Laplace transform solve
\[ mx'' + cx'+kx =0,\quad x(0)=a, \quad x'(0)=b.\]
where \(m>0,c>0,k>0\), and \(c^2-4km<0\) (system is underdamped).
Exercise \(\PageIndex{6.2.5}\)
Using the Laplace transform solve
\[ mx'' + cx'+kx =0,\quad x(0)=a, \quad x'(0)=b.\]
where \(m>0,c>0,k>0\), and \(c^2=4km\) (system is critically damped).
Exercise \(\PageIndex{6.2.6}\)
Solve \(x''+x=u(t-1)\) for initial conditions \(x(0)=0\) and \(x'(0)=0\).
Exercise \(\PageIndex{6.2.7}\)
Show the differentiation of the transform property. Suppose \(\mathcal{L}\{f(t)\}=F(s)\), then show
\[ \mathcal{L}\{-tf(t)\}=F'(s).\]
Hint: Differentiate under the integral sign.
Exercise \(\PageIndex{6.2.8}\)
Solve \(x'''+x=t^3u(t-1)\) for initial conditions \(x(0)=1\) and \(x'(0)=0\), \(x''(0)=0\).
Exercise \(\PageIndex{6.2.9}\)
Show the second shifting property: \( \mathcal{L}\{f(t-a)u(t-a)\}=e^{-as}\mathcal{L}\{f(t)\} \).
Exercise \(\PageIndex{6.2.10}\)
Let us think of the mass-spring system with a rocket from Example 6.2.2. We noticed that the solution kept oscillating after the rocket stopped running. The amplitude of the oscillation depends on the time that the rocket was fired (for 4 seconds in the example).
- Find a formula for the amplitude of the resulting oscillation in terms of the amount of time the rocket is fired.
- Is there a nonzero time (if so what is it?) for which the rocket fires and the resulting oscillation has amplitude 0 (the mass is not moving)?
Exercise \(\PageIndex{6.2.11}\)
Define
\[ f(t)= \left\{ \begin{array}{ccc} (t-1)^2 & if~1 \leq t<2, \\ 3-t & if~2 \leq t<3, \\ 0 & otherwise. \end{array} \right. \]
- Sketch the graph of \(f(t)\).
- Write down \(f(t)\) using the Heaviside function.
- Solve \(x''+x=f(t), x(0)=0,x'(0)=0\) using Laplace transform.
Exercise \(\PageIndex{6.2.12}\)
Find the transfer function for \(mx'' + cx'+kx =f(t)\) (assuming the initial conditions are zero).
Exercise \(\PageIndex{6.2.13}\)
Using the Heaviside function \(u(t)\), write down the function
\[ f(t)= \left\{ \begin{array}{ccc} 0 & if~~~~~t<1, \\ t-1 & if~1 \leq t<2, \\ if~~~~~2 \leq t. \end{array} \right. \]
- Answer
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\(f(t)=(t-1)(u(t-1)-u(t-2))+u(t-2)\)
Exercise \(\PageIndex{6.2.14}\)
Solve \(x''-x=(t^2-1)u(t-1)\) for initial conditions \(x(0)=1,x'(0)=2\) using the Laplace transform.
- Answer
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\(x(t)=(2e^{t-1}-t^{2}-1)u(t-1)-\frac{1}{2}e^{-t}+\frac{3}{2}e^{t}\)
Exercise \(\PageIndex{6.2.15}\)
Find the transfer function for \(x'+x=f(t)\) (assuming the initial conditions are zero).
- Answer
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\(H(s)=\frac{1}{s+1}\)
6.3: Convolution
Exercise \(\PageIndex{6.3.1}\)
Let \(f(t)=t^2\) for \(t \geq 0\), and \(g(t)=u(t-1)\). Compute \(f * g\).
Exercise \(\PageIndex{6.3.2}\)
Let \(f(t)=t\) for \(t \geq 0\), and \(g(t)=\sin t\) for \(t \geq 0\). Compute \(f * g\).
Exercise \(\PageIndex{6.3.3}\)
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 \(\PageIndex{6.3.4}\)
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.
Exercise \(\PageIndex{6.3.5}\)
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 \(\PageIndex{6.3.6}\)
Solve
\( x(t)=e^{-t} +\int_0^t\cos(t-\tau)x(\tau)~d\tau . \)
Exercise \(\PageIndex{6.3.7}\)
Solve
\( x(t)=\cos t +\int_0^t\cos(t-\tau)x(\tau)~d\tau . \)
Exercise \(\PageIndex{6.3.8}\)
Compute \(\mathcal{L}^{-1} \left\{ \frac{s}{(s^2+4)^2}\right\}\) using convolution.
Exercise \(\PageIndex{6.3.9}\)
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 \(\PageIndex{6.3.10}\)
Let \(f(t)=\cos t\) for \(t \geq 0\), and \(g(t)=e^{-t}\). Compute \(f * g\).
- Answer
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\(\frac{1}{2}(\cos t+\sin t-e^{-t})\)
Exercise \(\PageIndex{6.3.11}\)
Compute \(\mathcal{L}^{-1} \left\{ \frac{5}{s^4+s^2}\right\}\) using convolution.
- Answer
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\(5t-5\sin t\)
Exercise \(\PageIndex{6.3.12}\)
Solve \(x''+x=\sin t, x(0)=0, x'(0)=0\) using convolution.
- Answer
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\(\frac{1}{2}(\sin t-t\cos t)\)
Exercise \(\PageIndex{6.3.13}\)
Solve \(x'''+x'=f(t), x(0)=0, x'(0)=0,x''(0)=0\) using convolution. Write the result as a definite integral.
- Answer
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\(\int_{0}^{t} f(\tau )(1-\cos (t-\tau ))d\tau \)
6.4: Dirac delta and impulse response
Exercise \(\PageIndex{6.4.1}\)
Solve (find the impulse response) \( x'' + x' + x = \delta(t),x(0) = 0, x'(0)=0.\)
Exercise \(\PageIndex{6.4.2}\)
Solve (find the impulse response) \(x'' + 2 x' + x = \delta(t), x(0) = 0, x'(0)=0.\)
Exercise \(\PageIndex{6.4.3}\)
A pulse can come later and can be bigger. Solve \(x'' + 4 x = 4\delta(t-1), x(0) = 0, x'(0)=0.\)
Exercise \(\PageIndex{6.4.4}\)
Suppose that \(f(t)\) and \(g(t)\) are differentiable functions and suppose that \(f(t) = g(t) = 0\) for all \(t \leq 0\). Show that \[ (f * g)'(t) = (f' * g)(t) = (f * g')(t) .\]
Exercise \(\PageIndex{6.4.5}\)
Suppose that \(L x = \delta(t), x(0) = 0, x'(0) = 0\), has the solution \(x = e^{-t}\) for \(t>0\). Find the solution to \(Lx = t^2, x(0) = 0, x'(0) = 0\) for \(t > 0\).
Exercise \(\PageIndex{6.4.6}\)
Compute \(\mathcal{L}^{-1} \left\{ \frac{s^2+s+1}{s^2} \right\}\).
Exercise \(\PageIndex{6.4.7}\): (challenging)
Solve Example 6.4.3 via integrating 4 times in the \(x\) variable.
Exercise \(\PageIndex{6.4.8}\)
Suppose we have a beam of length \(1\) simply supported at the ends and suppose that force \(F=1\) is applied at \(x=\frac{3}{4}\) in the downward direction. Suppose that \(EI=1\) for simplicity. Find the beam deﬂection \(y(x)\).
Exercise \(\PageIndex{6.4.9}\)
Solve (find the impulse response) \(x'' = \delta(t),\: x(0) = 0,\: x'(0)=0\).
- Answer
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\(x(t)=t\)
Exercise \(\PageIndex{6.4.10}\)
Solve (find the impulse response) \(x' + a x = \delta(t),\: x(0) = 0,\: x'(0)=0\).
- Answer
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\(x(t)=e^{-at}\)
Exercise \(\PageIndex{6.4.11}\)
Suppose that \(L x = \delta(t), x(0) = 0, x'(0) = 0\), has the solution \(x(t) = \cos(t)\) for \(t>0\). Find (in closed form) the solution to \(Lx = \sin(t), x(0) = 0, x'(0) = 0 for t > 0\).
- Answer
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\(x(t)=(\cos\ast\sin )(t)=\frac{1}{2}t\sin (t)\)
Exercise \(\PageIndex{6.4.12}\)
Compute \({\mathcal{L}}^{-1} \left\{ \frac{s^2}{s^2+1} \right\}\).
- Answer
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\(\delta (t)-\sin (t)\)
Exercise \(\PageIndex{6.4.13}\)
Compute \({\mathcal{L}}^{-1} \left\{ \frac{3 s^2 e^{-s} + 2}{s^2} \right\}\).
- Answer
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\(3\delta (t-1)+2t\)
6.5: Solving PDEs with the Laplace Transform
Exercise \(\PageIndex{6.5.1}\)
Solve \[\begin{aligned} & y_t + y_x = 1, \qquad 0 < x < \infty, \enspace t > 0, \\ & y(0,t) = 1, \quad y(x,0) = 0 .\end{aligned}\]
Exercise \(\PageIndex{6.5.2}\)
Solve \[\begin{aligned} & y_t + \alpha y_x = 0, \qquad 0 < x < \infty, \enspace t > 0, \\ & y(0,t) = t, \quad y(x,0) = 0 .\end{aligned}\]
Exercise \(\PageIndex{6.5.3}\)
Solve \[\begin{aligned} & y_t + 2 y_x = x+t, \qquad 0 < x < \infty, \enspace t > 0, \\ & y(0,t) = 0, \quad y(x,0) = 0 .\end{aligned}\]
Exercise \(\PageIndex{6.5.4}\)
For an \(\alpha > 0\) , solve \[\begin{aligned} & y_t + \alpha y_x + y = 0, \qquad 0 < x < \infty, \enspace t > 0, \\ & y(0,t) = \sin(t), \quad y(x,0) = 0 .\end{aligned}\]
Exercise \(\PageIndex{6.5.5}\)
Find the corresponding ODE problem for \(Y(x)\) , after transforming the \(t\) variable \[\begin{aligned} & y_{tt} + 3y_{xx} + y_{xt} + 3 y_x + y = \sin(x) + t, \qquad 0 < x < 1, \enspace t > 0, \\ & y(0,t) = 1, \quad y(1,t) = t, \quad y(x,0) = 1-x, \quad y_t(x,0) = 1 .\end{aligned}\] Do not solve the problem.
Exercise \(\PageIndex{6.5.6}\)
Write down a solution to \[\begin{aligned} & y_t = y_{xx}, \qquad 0 < x < \infty, \enspace t > 0,\\ & y_x(0,t) = e^{-t}, \quad y(x,0) = 0 ,\end{aligned}\] as an definite integral (convolution).
Exercise \(\PageIndex{6.5.7}\)
Use the Laplace transform in \(t\) to solve \[\begin{aligned} & y_{tt} = y_{xx}, \qquad -\infty < x < \infty, \enspace t > 0,\\ & y_t(x,0) = \sin(x), \quad y(x,0) = 0 .\end{aligned}\] Hint: Note that \(e^{sx}\) does not go to zero as \(s \to \infty\) for positive \(x\) , and \(e^{-sx}\) does not go to zero as \(s \to \infty\) for negative \(x\) .
Exercise \(\PageIndex{6.5.8}\)
Solve \[\begin{aligned} & y_t + y_x = 1, \qquad 0 < x < \infty, \enspace t > 0, \\ & y(0,t) = 0, \quad y(x,0) = 0 .\end{aligned}\]
- Answer
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\(y=(x-t)u(t-x)+t\)
Exercise \(\PageIndex{6.5.9}\)
For a \(c > 0\) , solve \[\begin{aligned} & y_t + y_x + c y = 0, \qquad 0 < x < \infty, \enspace t > 0, \\ & y(0,t) = \sin(t), \quad y(x,0) = 0 .\end{aligned}\]
- Answer
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\(y=e^{-cx}\sin (t-x)u(t-x)\)
Exercise \(\PageIndex{6.5.10}\)
Find the corresponding ODE problem for \(Y(x)\) , after transforming the \(t\) variable \[\begin{aligned} & y_{tt} + 3y_{xx} + y = x+t, \qquad -1 < x < 1, \enspace t > 0, \\ & y(-1,t) = 0, \quad y(1,t) = 0, \quad y(x,0) = (1-x^2) , \quad y_t(x,0) = 0.\end{aligned}\] Do not solve the problem.
- Answer
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\(s^{2}Y(x)-s(1+x^{2})+3Y''(x)+Y(x)=\frac{x}{s}+\frac{1}{s^{2}},\quad Y(-1)=0,\quad Y(1)=0.\)
Exercise \(\PageIndex{6.5.11}\)
Use the Laplace transform in \(t\) to solve \[\begin{aligned} & y_{tt} = y_{xx}, \qquad -\infty < x < \infty, \enspace t > 0,\\ & y_t(x,0) = x^2, \quad y(x,0) = 0 .\end{aligned}\] Hint: Note that \(e^{sx}\) does not go to zero as \(s \to \infty\) for positive \(x\) , and \(e^{-sx}\) does not go to zero as \(s \to \infty\) for negative \(x\) .
- Answer
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\(y=tx^{2}+\frac{t^{3}}{3}\)