6.5 Practice Problems
Problem 6.5.1. Compute directly from the definition: \(\mathcal {L}\{1\}\), \(\mathcal {L}\{e^{at}\}\), \(\mathcal {L}\{t\}\) and \(\mathcal {L}\{\sin bt\}\), stating in each case the values of \(s\) for which the integral converges. Where to start: for \(\sin bt\), integrate by parts twice and solve for the transform, or write \(\sin bt\) in exponential form and use linearity.
Problem 6.5.2. Show that \(\mathcal {L}\{t^{1/2}\} = \dfrac {\sqrt {\pi }}{2s^{3/2}}\). Where to start: Example 6.1.5 with \(a=\tfrac 12\), then \(\Gamma (3/2)=\tfrac 12\Gamma (1/2)\) from the functional equation.
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Solution. By Example 6.1.5, \(\mathcal {L}\{t^{1/2}\} = \Gamma (3/2)/s^{3/2}\). The functional equation gives \(\Gamma (3/2) = \tfrac 12\Gamma (1/2) = \tfrac 12\sqrt {\pi }\), whence the result. Note that no elementary method produces this: the transform of a half-power requires the gamma function.
Problem 6.5.4. Solve \(y'' + y = \delta (t-\pi )\) with \(y(0)=y'(0)=0\), and describe the motion in words. What does the delta function represent physically? Where to start: the transform of \(\delta (t-a)\) is \(e^{-as}\); the answer should be zero until \(t=\pi \) and sinusoidal thereafter.
Problem 6.5.5. Prove the first shifting theorem \(\mathcal {L}\{e^{at}f(t)\}=F(s-a)\) directly from the definition, and use it to find \(\mathcal {L}\{e^{-2t}\cos 3t\}\).
Problem 6.5.6. Prove the convolution theorem, Theorem 6.2.5, by writing the product \(F(s)G(s)\) as a double integral and changing the order of integration. State the region in the \((\tau ,t)\) plane over which the integration runs. Where to start: the substitution \(t = \tau + u\) turns the product of two integrals into an integral over the wedge \(0\leq \tau \leq t\).
Problem 6.5.7. Use the convolution theorem to invert \(\dfrac {1}{s^{2}\left (s^{2}+1\right )}\), and check the answer by partial fractions.
Problem 6.5.8. Show that \(\mathcal {L}\left \{\dfrac {\sin t}{t}\right \} = \arctan \left (1/s\right )\). Where to start: use the property \(\mathcal {L}\{f(t)/t\} = \int ^{\infty }_{s}F(u)\,du\), itself proved by integrating the defining integral with respect to \(s\).
Problem 6.5.9. Explain why \(F(s) = s/(s+1)\) cannot be the Laplace transform of any piecewise continuous function of exponential order. Where to start: Theorem 6.1.3.
Problem 6.5.10. The gamma function was defined in Chapter 2 by an integral converging only for \(s>0\). Show that \(\mathcal {L}\{t^{a}\}\) exists for exactly the same range of \(a\), and explain why the two conditions coincide.
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