9.12: Problems
- Page ID
- 90981
In this problem you will show that the sequence of functions \[f_{n}(x)=\frac{n}{\pi}\left(\frac{1}{1+n^{2} x^{2}}\right)\nonumber \] approaches \(\delta(x)\) as \(n \rightarrow \infty\). Use the following to support your argument:
- Show that \(\lim _{n \rightarrow \infty} f_{n}(x)=0\) for \(x \neq 0\).
- Show that the area under each function is one.
Verify that the sequence of functions \(\left\{f_{n}(x)\right\}_{n=1}^{\infty}\), defined by \(f_{n}(x)=\) \(\frac{n}{2} e^{-n|x|}\), approaches a delta function.
Evaluate the following integrals:
- \(\int_{0}^{\pi} \sin x \delta\left(x-\frac{\pi}{2}\right) d x\).
- \(\int_{-\infty}^{\infty} \delta\left(\frac{x-5}{3} e^{2 x}\right)\left(3 x^{2}-7 x+2\right) d x\).
- \(\int_{0}^{\pi} x^{2} \delta\left(x+\frac{\pi}{2}\right) d x\).
- \(\int_{0}^{\infty} e^{-2 x} \delta\left(x^{2}-5 x+6\right) d x\). [See Problem 4.]
- \(\int_{-\infty}^{\infty}\left(x^{2}-2 x+3\right) \delta\left(x^{2}-9\right) d x\). [See Problem 4.]
For the case that a function has multiple roots, \(f\left(x_{i}\right)=0, i=1,2, \ldots\), it can be shown that \[\delta(f(x))=\sum_{i=1}^{n} \frac{\delta\left(x-x_{i}\right)}{\left|f^{\prime}\left(x_{i}\right)\right|} .\nonumber \] Use this result to evaluate \(\int_{-\infty}^{\infty} \delta\left(x^{2}-5 x-6\right)\left(3 x^{2}-7 x+2\right) d x\).
Find a Fourier series representation of the Dirac delta function, \(\delta(x)\), on \([-L, L]\).
For \(a>0\), find the Fourier transform, \(\hat{f}(k)\), of \(f(x)=e^{-a|x|}\).
Use the result from the last problem plus properties of the Fourier transform to find the Fourier transform, of \(f(x)=x^{2} e^{-a|x|}\) for \(a>0\).
Find the Fourier transform, \(\hat{f}(k)\), of \(f(x)=e^{-2 x^{2}+x}\).
Prove the second shift property in the form \[F\left[e^{i \beta x} f(x)\right]=\hat{f}(k+\beta) .\nonumber \]
A damped harmonic oscillator is given by \[f(t)=\left\{\begin{array}{cl} A e^{-\alpha t} e^{i \omega_{0} t}, & t \geq 0, \\ 0, & t<0 . \end{array}\right.\nonumber \]
- Find \(\hat{f}(\omega)\) and
- the frequency distribution \(|\hat{f}(\omega)|^{2}\).
- Sketch the frequency distribution.
Show that the convolution operation is associative: \((f *(g * h))(t)=\) \(((f * g) * h)(t)\).
In this problem you will directly compute the convolution of two Gaussian functions in two steps.
- Use completing the square to evaluate \[\int_{-\infty}^{\infty} e^{-\alpha t^{2}+\beta t} d t .\nonumber \]
- Use the result from part a to directly compute the convolution in Example 9.6.6: \[(f * g)(x)=e^{-b x^{2}} \int_{-\infty}^{\infty} e^{-(a+b) t^{2}+2 b x t} d t .\nonumber \]
You will compute the (Fourier) convolution of two box functions of the same width. Recall the box function is given by \[f_{a}(x)= \begin{cases}1, & |x| \leq a \\ 0, & |x|>a .\end{cases}\nonumber \] Consider \(\left(f_{a} * f_{a}\right)(x)\) for different intervals of \(x\). A few preliminary sketches would help. In Figure \(\PageIndex{1}\) the factors in the convolution integrand are show for one value of \(x\). The integrand is the product of the first two functions. The convolution at \(x\) is the area of the overlap in the third figure. Think about how these pictures change as you vary \(x\). Plot the resulting areas as a function of \(x\). This is the graph of the desired convolution.
Define the integrals \(I_{n}=\int_{-\infty}^{\infty} x^{2 n} e^{-x^{2}} d x\). Noting that \(I_{0}=\sqrt{\pi}\),
- Find a recursive relation between \(I_{n}\) and \(I_{n-1}\).
- Use this relation to determine \(I_{1}, I_{2}\) and \(I_{3}\).
- Find an expression in terms of \(n\) for \(I_{n}\).
Find the Laplace transform of the following functions.
- \(f(t)=9 t^{2}-7\).
- \(f(t)=e^{5 t-3}\).
- \(f(t)=\cos 7 t\).
- \(f(t)=e^{4 t} \sin 2 t\).
- \(f(t)=e^{2 t}(t+\cosh t)\).
- \(f(t)=t^{2} H(t-1)\).
- \(f(t)=\left\{\begin{array}{cc}\sin t, & t<4 \pi \\ \sin t+\cos t, & t>4 \pi\end{array} .\right.\)
- \(f(t)=\int_{0}^{t}(t-u)^{2} \sin u d u\).
- \(f(t)=(t+5)^{2}+t e^{2 t} \cos 3 t\) and write the answer in the simplest form.
Find the inverse Laplace transform of the following functions using the properties of Laplace transforms and the table of Laplace transform pairs.
- \(F(s)=\frac{18}{s^{3}}+\frac{7}{s} .\)
- \(F(s)=\frac{1}{s-5}-\frac{2}{s^{2}+4} .\)
- \(F(s)=\frac{s+1}{s^{2}+1} .\)
- \(F(s)=\frac{3}{s^{2}+2 s+2} .\)
- \(F(s)=\frac{1}{(s-1)^{2}} .\)
- \(F(s)=\frac{e^{-3 s}}{s^{2}-1} .\)
- \(F(s)=\frac{1}{s^{2}+4 s-5} .\)
- \(F(s)=\frac{s+3}{s^{2}+8 s+17} .\)
Compute the convolution \((f * g)(t)\) (in the Laplace transform sense) and its corresponding Laplace transform \(\mathcal{L}[f * g]\) for the following functions:
- \(f(t)=t^{2}, g(t)=t^{3} \text {. }\)
- \(f(t)=t^{2}, g(t)=\cos 2 t \text {. }\)
- \(f(t)=3t^2-2t+1, g(t)=e^{-3t}\text{.}\)
- \(f(t)=\delta (t-\frac{\pi}{4}), g(t)=\sin 5t\text{.}\)
For the following problems draw the given function and find the Laplace transform in closed form.
- \(f(t)=1+\sum_{n=1}^{\infty}(-1)^{n} H(t-n)\).
- \(f(t)=\sum_{n=0}^{\infty}[H(t-2 n+1)-H(t-2 n)]\).
- \(f(t)=\sum_{n=0}^{\infty}(t-2 n)[H(t-2 n)-H(t-2 n-1)]+(2 n+2-t)[H(t-\) \(2 n-1)-H(t-2 n-2)]\).
Use the convolution theorem to compute the inverse transform of the following:
- \(F(s)=\frac{2}{s^{2}\left(s^{2}+1\right)} .\)
- \(F(s)=\frac{e^{-3 s}}{s^{2}} .\)
- \(F(s)=\frac{1}{s\left(s^{2}+2 s+5\right)} .\)
Find the inverse Laplace transform two different ways: i) Use Tables. ii) Use the Bromwich Integral.
- \(F(s)=\frac{1}{s^{3}(s+4)^{2}} .\)
- \(F(s)=\frac{1}{s^{2}-4 s-5} .\)
- \(F(s)=\frac{s+3}{s^{2}+8 s+17} .\)
- \(F(s)=\frac{s+1}{(s-2)^{2}(s+4)} .\)
- \(F(s)=\frac{s^{2}+8 s-3}{\left(s^{2}+2 s+1\right)\left(s^{2}+1\right)} .\)
Use Laplace transforms to solve the following initial value problems. Where possible, describe the solution behavior in terms of oscillation and decay.
- \(y^{\prime \prime}-5 y^{\prime}+6 y=0, \: y(0)=2, \: y^{\prime}(0)=0 .\)
- \(y^{\prime \prime}-y=t e^{2 t}, \: y(0)=0, \: y^{\prime}(0)=1 .\)
- \(y^{\prime \prime}+4 y=\delta(t-1), \: y(0)=3, \: y^{\prime}(0)=0 .\)
- \(y^{\prime \prime}+6 y^{\prime}+18 y=2 H(\pi-t), \:y(0)=0,\: y^{\prime}(0)=0 .\)
Use Laplace transforms to convert the following system of differential equations into an algebraic system and find the solution of the differential equations. \[\begin{array}{lll} x^{\prime \prime} & =3 x-6 y, & x(0) & =1, & x^{\prime}(0) & =0 \\ y^{\prime \prime} & =x+y, & & y(0) & =0, & y^{\prime}(0) & =0 . \end{array}\nonumber \]
Use Laplace transforms to convert the following nonhomogeneous systems of differential equations into an algebraic system and find the solutions of the differential equations.
- \[\begin{aligned} &x^{\prime}=2 x+3 y+2 \sin 2 t, \quad x(0)=1, \\ &y^{\prime}=-3 x+2 y, \quad y(0)=0 . \end{aligned}\]
- \[\begin{aligned} &x^{\prime}=-4 x-y+e^{-t}, \quad x(0)=2, \\ &y^{\prime}=x-2 y+2 e^{-3 t}, \quad y(0)=-1 . \end{aligned}\]
- \[\begin{aligned} &x^{\prime}=x-y+2 \cos t, \quad x(0)=3, \\ &y^{\prime}=x+y-3 \sin t, \quad y(0)=2 . \end{aligned}\]
Consider the series circuit in Problem 2.8.20 and in Figure ?? with \(L=\) \(1.00 \mathrm{H}, R=1.00 \times 10^{2} \Omega, C=1.00 \times 10^{-4} \mathrm{~F}\), and \(V_{0}=1.00 \times 10^{3} \mathrm{~V}\).
- Write the second order differential equation for this circuit.
- Suppose that no charge is present and no current is flowing at time \(t=0\) when \(V_{0}\) is applied. Use Laplace transforms to find the current and the charge on the capacitor as functions of time.
- Replace the battery with the alternating source \(V(t)=V_{0} \sin 2 \pi f t\) with \(V_{0}=1.00 \times 10^{3} \mathrm{~V}\) and \(f=150 \mathrm{~Hz}\). Again, suppose that no charge is present and no current is flowing at time \(t=0\) when the AC source is applied. Use Laplace transforms to find the current and the charge on the capacitor as functions of time.
- Plot your solutions and describe how the system behaves over time.
Use Laplace transforms to sum the following series.
- \(\sum_{n=0}^{\infty} \frac{(-1)^{n}}{1+2 n}\).
- \(\sum_{n=1}^{\infty} \frac{1}{n(n+3)}\).
- \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n(n+3)}\).
- \(\sum_{n=0}^{\infty} \frac{(-1)^{n}}{n^{2}-a^{2}}\).
- \(\sum_{n=0}^{\infty} \frac{1}{(2 n+1)^{2}-a^{2}}\).
- \(\sum_{n=1}^{\infty} \frac{1}{n} e^{-a n}\).
Use Laplace transforms to prove \[\sum_{n=1}^{\infty} \frac{1}{(n+a)(n+b)}=\frac{1}{b-a} \int_{0}^{1} \frac{u^{a}-u^{b}}{1-u} d u .\nonumber \] Use this result to evaluate the sums
- \(\sum_{n=1}^{\infty} \frac{1}{n(n+1)}\).
- \(\sum_{n=1}^{\infty} \frac{1}{(n+2)(n+3)}\).
Do the following.
- Find the first four nonvanishing terms of the Maclaurin series expansion of \(f(x)=\frac{x}{e^{x}-1}\).
- Use the result in part a. to determine the first four nonvanishing Bernoulli numbers, \(B_{n}\).
- Use these results to compute \(\zeta(2 n)\) for \(n=1,2,3,4\).
Given the following Laplace transforms, \(F(s)\), find the function \(f(t)\). Note that in each case there are an infinite number of poles, resulting in an infinite series representation.
- \(F(s)=\frac{1}{s^{2}\left(1+e^{-s}\right)}\).
- \(F(s)=\frac{1}{s \sinh s}\).
- \(F(s)=\frac{\sinh s}{s^{2} \cosh s}\).
- \(F(s)=\frac{\sinh (\beta \sqrt{s} x)}{s \sinh (\beta \sqrt{s} L)}\).
Consider the initial boundary value problem for the heat equation: \[\begin{array}{cc} u_{t}=2 u_{x x}, & 0<t, \quad 0 \leq x \leq 1, \\ u(x, 0)=x(1-x), & 0<x<1, \\ u(0, t)=0, & t>0, \\ u(1, t)=0, & t>0 . \end{array}\nonumber \] Use the finite transform method to solve this problem. Namely, assume that the solution takes the form \(u(x, t)=\sum_{n=1}^{\infty} b_{n}(t) \sin n \pi x\) and obtain an ordinary differential equation for \(b_{n}\) and solve for the \(b_{n}\) ’s for each \(n\).