Chapter 10

Sketch the given function and determine whether it is piecewise continuous on \([0, \infty)\). $$f(t)=\frac{1}{t-2}$$

Solve the given initial-value problem. $$y^{\prime}+2 y=u_{\pi}(t) \sin 2 t, \quad y(0)=3$$.

Solve the given initial-value problem up to the evaluation of a convolution integral. \(y^{\prime \prime}-a^{2} y=f(t), \quad y(0)=\alpha, \quad y^{\prime}(0)=\beta,\) where \(a, \alpha,\) and \(\beta\) are constants and \(a \neq 0\)

Determine \(L^{-1}[F]\). $$F(s)=\frac{2}{\sqrt{s+3}}$$.

The current \(i(t)\) in an RL circuit is governed by the differential equation $$\frac{d i}{d t}+\frac{R}{L} i=\frac{1}{L} E(t)$$ where \(R\) and \(L\) are constants. (a) Use the Laplace transform to determine \(i(t)\) if \(E(t)=E_{0},\) a constant. There is no current flowing initially. (b) Repeat part (a) in the case when \(E(t)=E_{0}\) \(\sin \omega t,\) where \(\omega\) is a constant. The Laplace transform can also be used to solve initial-value problems for systems of linear differential equations. The remaining problems deal with this.

Let \(f(x)=x e^{a x},\) where \(a\) is a constant. (a) Show that $$ L\left\\{f^{\prime}(x)\right\\}=a L\\{f(x)\\}+\frac{1}{s-a} $$ (b) Use the result from (a) together with the expression for the Laplace transform of the derivative of a function, to determine \(L\left\\{x e^{a x}\right\\}\) without integrating. (c) Use mathematical induction to establish that \(L\left\\{x^{n} e^{a x}\right\\}=\frac{n !}{(s-a)^{n+1}}, \quad s>a, \quad n=1,2, \dots\)

Solve the given initial-value problem. \(y^{\prime}+3 y=f(t), \quad y(0)=1,\) where $$ f(t)=\left\\{\begin{array}{lr} 1, & 0 \leq t<1, \\ 0, & t \geq 1. \end{array}\right. $$

Solve the given initial-value problem up to the evaluation of a convolution integral. \(y^{\prime \prime}-(a+b) y^{\prime}+a b y=f(t), \quad y(0)=\alpha, \quad y^{\prime}(0)=\) \(\beta,\) where \(a, b, \alpha,\) and \(\beta\) are constants and \(a \neq b\)

Consider the initial-value problem $$\begin{array}{l}x_{1}^{\prime}=a_{11} x_{1}+a_{12} x_{2}+b_{1}(t) \\\x_{2}^{\prime}=a_{21} x_{1}+a_{22} x_{2}+b_{2}(t) \\\x_{1}(0)=\alpha_{1}, \quad x_{2}(0)=\alpha_{2}\end{array}$$ where the \(a_{i j}, \alpha_{1},\) and \(\alpha_{2}\) are constants. Show that the Laplace transforms of \(x_{1}(t)\) and \(x_{2}(t)\) must satisfy the linear system $$\begin{aligned}\left(s-a_{11}\right) X_{1}(s)-& a_{12} X_{2}(s)=\alpha_{1}+B_{1}(s) \\ -a_{12} X_{1}(s)+\left(s-a_{22}\right) X_{2}(s) &=\alpha_{2}+B_{2}(s)\end{aligned}$$ This system can be solved quite easily (for example, by Cramer's Rule) to determine \(X_{1}(s)\) and \(X_{2}(s),\) and then \(x_{1}(t)\) and \(x_{2}(t)\) can be obtained by taking the inverse Laplace transform.

Determine \(L^{-1}[F]\). $$F(s)=\frac{2}{(s-1)^{2}+4}$$.

Sketch the given function and determine its Laplace transform. $$f(t)=\left\\{\begin{aligned} 1, & 0 \leq t \leq 2 \\ -1, & t > 2 \end{aligned}\right.$$

Let \(y(t)\) be the solution to the initial-value problem \(y^{\prime}+a y=f(t), y(0)=y_{0},\) where \(a\) and \(y_{0}\) are constants. Verify that $$ L[y]=\frac{L[f]}{s+a}+\frac{y_{0}}{s+a} $$ and show that $$ y(t)=y_{0} e^{-a t}+\int_{0}^{t} e^{-a(t-w)} f(w) d w $$

Solve the given initial-value problem. $$\begin{aligned} y^{\prime}-3 y=& f(t), \quad y(0)=2, \text { where } \\ & f(t)=\left\\{\begin{array}{cc} \sin t, & 0 \leq t<\pi / 2, \\ 1, & t \geq \pi / 2. \end{array}\right. \end{aligned}$$

Determine \(L^{-1}[F]\). $$F(s)=\frac{s+2}{(s+2)^{2}+9}$$.

Sketch the given function and determine its Laplace transform. $$f(t)=\left\\{\begin{array}{lr} t, & 0 \leq t \leq 1 \\ 0, & t \geq 1 \end{array}\right.$$

Solve the given initial-value problem up to the evaluation of a convolution integral. \(y^{\prime \prime}-2 a y^{\prime}+\left(a^{2}+b^{2}\right) y=f(t), \quad y(0)=\alpha, \quad y^{\prime}(0)=\) \(\beta,\) where \(a, b, \alpha,\) and \(\beta\) are constants, and \(b \neq 0\)

Solve the given initial-value problem. \(x_{1}^{\prime}=-4 x_{1}-2 x_{2}, \quad x_{2}^{\prime}=x_{1}-x_{2}\) \(x_{1}(0)=0, x_{2}(0)=1\).

Solve the given initial-value problem. \(x_{1}^{\prime}=-3 x_{1}+4 x_{2}, \quad x_{2}^{\prime}=-x_{1}+2 x_{2}\) \(x_{1}(0)=2, x_{2}(0)=1\).

Determine \(L^{-1}[F]\). $$F(s)=\frac{s}{(s-3)^{2}+4}$$.

Sketch the given function and determine its Laplace transform. $$f(t)=\left\\{\begin{array}{cc} t, & 0 \leq t < 1 \\ 1, & 1 \leq t < 3 \\ e^{t-3}, & t > 3 \end{array}\right.$$

Show that the general solution to the initial-value problem $$ \begin{array}{c} y^{(n)}+a_{1} y^{(n-1)}+\cdots+a_{n} y=f(t) \\ y(0)=0, y^{\prime}(0)=0, \ldots, y^{(n-1)}(0)=0 \end{array} $$ is $$ y(t)=\int_{0}^{t} K(t-w) f(w) d w $$ for an appropriate function \(K(t)\) that should be determined.

Solve the given initial-value problem. \(y^{\prime}-3 y=10 e^{-(t-a)} \sin [2(t-a)] u_{a}(t), \quad y(0)=5\) where \(a\) is a positive constant.

Solve the given Volterra integral equation. $$x(t)=e^{-t}+4 \int_{0}^{t}(t-\tau) x(\tau) d \tau$$

Solve the given initial-value problem. $$y^{\prime \prime}-y=u_{1}(t), \quad y(0)=2, \quad y^{\prime}(0)=0$$.

Solve the given Volterra integral equation. $$x(t)=2 e^{3 t}-\int_{0}^{t} e^{2(t-\tau)} x(\tau) d \tau$$

Establish the formula for \(L\left[f^{(n)}\right],\) the Laplace transform of the \(n\) th derivative of \(f,\) given in the text. [Hint: Use induction on \(n .]\)

Determine \(L^{-1}[F]\). $$F(s)=\frac{5}{(s-2)^{2}+16}$$.

Sketch the given function and determine its Laplace transform. $$f(t)=\left\\{\begin{array}{lr} 0, & 0 \leq t \leq 1 \\ t, & 1 2 \end{array}\right.$$

Use the Laplace transform to solve the given initial-value problem. $$y^{\prime \prime}-3 y^{\prime}-4 y=4 e^{-t}, \quad y(0)=1, \quad y^{\prime}(0)=1$$

Use the Laplace transform to solve the given initial-value problem. $$y^{\prime \prime}-2 y^{\prime}-8 y=5, \quad y(0)=1, \quad y^{\prime}(0)=0$$

Solve the given initial-value problem. $$y^{\prime \prime}-y^{\prime}-2 y=1-3 u_{2}(t), \quad y(0)=1, \quad y^{\prime}(0)=-2$$.

Solve the given Volterra integral equation. $$x(t)=4 e^{t}+3 \int_{0}^{t} e^{-(t-\tau)} x(\tau) d \tau$$

Recall that according to Euler's formula $$ e^{i b t}=\cos b t+i \sin b t $$ since the Laplace transform is linear, it follows that $$ \begin{aligned} L[\cos b t] &=\operatorname{Re}\left(L\left[e^{i b t}\right]\right) \\ L[\sin b t] &=\operatorname{Im}\left(L\left[e^{i b t}\right]\right) \end{aligned} $$ Find \(\left.L\left[e^{i b t}\right], \text { and hence, derive Equations ( } 10.1 .4\right)\) and \((10.1 .5)\)

Determine \(L^{-1}[F]\). $$F(s)=\frac{6}{s^{2}+2 s+2}$$.

Use the Laplace transform to solve the given initial-value problem. $$y^{\prime \prime}+9 y=8 \cos 3 t, \quad y(0)=1, \quad y^{\prime}(0)=0$$

Solve the given initial-value problem. $$y^{\prime \prime}-4 y=u_{1}(t)-u_{2}(t), \quad y(0)=0, \quad y^{\prime}(0)=4$$.

Solve the given Volterra integral equation. $$x(t)=1+2 \int_{0}^{t} \sin (t-\tau) x(\tau) d \tau$$

Determine \(L^{-1}[F]\). $$F(s)=\frac{s-2}{s^{2}+2 s+26}$$.

Use the Laplace transform to solve the given initial-value problem. $$y^{\prime \prime}+y=f(t), \quad y(0)=0, \quad y^{\prime}(0)=1, \text { where }$$ $$f(t)=\left\\{\begin{array}{lr} 1, & 0 \leq t < \pi / 2 \\ 0, & t \geq \pi / 2 \end{array}\right.$$

Solve the given initial-value problem. $$y^{\prime \prime}+y=t-u_{1}(t)(t-1), \quad y(0)=2, \quad y^{\prime}(0)=1$$.

Solve the given Volterra integral equation. $$x(t)=e^{2 t}+5 \int_{0}^{t} \cos [2(t-\tau)] x(\tau) d \tau$$

Determine \(L^{-1}[F]\). $$F(s)=\frac{2 s}{s^{2}-4 s+13}$$.

Use mathematical induction to prove that for every positive integer \(n\) $$ L\left[t^{n}\right]=\frac{n !}{s^{n+1}} $$

$$y^{\prime \prime}+4 y=4 \sin t+3 \delta(t-2), \quad y(0)=2, \quad y^{\prime}(0)=1$$

Solve the given Volterra integral equation. $$x(t)=2\left\\{1+\int_{0}^{t} \cos [2(t-\tau)] x(\tau) d \tau\right\\}$$

Solve the given initial-value problem. $$\begin{aligned} &y^{\prime \prime}+3 y^{\prime}+2 y=10 u_{\pi / 4}(t) \sin (t-\pi / 4)\\\ &y(0)=1, \quad y^{\prime}(0)=0 \end{aligned}$$.

Determine \(L^{-1}[F]\). $$F(s)=\frac{s}{(s+1)^{2}+4}$$.

(a) By making the change of variables \(t=\frac{x^{2}}{s}(s > 0)\) in the integral that defines the Laplace transform, show that $$ L\left[t^{-1 / 2}\right]=2 s^{-1 / 2} \int_{0}^{\infty} e^{-x^{2}} d x $$ (b) Use your result in (a) to show that $$ \left(L\left[t^{-1 / 2}\right]\right)^{2}=4 s^{-1} \int_{0}^{\infty} \int_{0}^{\infty} e^{-\left(x^{2}+y^{2}\right)} d x d y $$ (c) By changing to polar coordinates, evaluate the double integral in (b), and hence, show that $$ L\left[t^{-1 / 2}\right]=\sqrt{\frac{\pi}{s}}, s > 0 $$

Use the Laplace transform to solve the given initial-value problem. $$y^{\prime \prime}+2 y^{\prime}+y=\delta(t-4), \quad y(0)=0, \quad y^{\prime}(0)=0$$

Show that the initial-value problem $$ y^{\prime \prime}+y=f(t), \quad y(0)=0, \quad y^{\prime}(0)=0 $$ can be reformulated as the integral equation $$ x(t)=f(t)-\int_{0}^{t}(t-\tau) x(\tau) d \tau $$ where \(y^{\prime \prime}(t)=x(t)\)