Chapter 10

Determine the Laplace transform of \(f\). $$f(t)=3 t e^{-t}$$.

Determine a function \(f(t)\) that has the given Laplace transform \(F(s)\). $$F(s)=\frac{s-2}{s^{2}+2 s+2}$$

Express \(L^{-1}[F(s) G(s)]\) in terms of a convolution integral. $$F(s)=\frac{s+1}{s^{2}+2 s+2}, \quad G(s)=\frac{1}{(s+3)^{2}}$$

Determine the Laplace transform of \(f\). $$f(t)=t^{3} e^{-4 t}$$.

Determine the inverse Laplace transform of \(F.\) $$F(s)=\frac{e^{-2 s}}{(s-3)^{3}}$$.

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}+5 y^{\prime}+4 y=20 \sin 2 t, \quad y(0)=-1, \quad y^{\prime}(0)=2\).

This exercise verifies the claim in the text that the Laplace transform defines a one-to-one linear transformation from \(V\) to \(\operatorname{Rng}(L) .\) Let \(f\) be a continuous function of exponential order. It suffices to prove that if \(f\) is not identically zero, then \(L[f] \neq 0\). [Hint: Show that \(L[|f|]>0\) if \(f\) is not identically zero.]

Use the linearity of \(L\) and the formulas derived in this section to determine \(L[f]\). \(f(t)=4 \cos ^{2} b t,\) where \(b\) is constant.

Express \(L^{-1}[F(s) G(s)]\) in terms of a convolution integral. $$F(s)=\frac{2}{s^{2}+6 s+10}, \quad G(s)=\frac{2}{s-4}$$

Determine the Laplace transform of \(f\). $$f(t)=e^{t}-t e^{-2 t}$$.

Determine the inverse Laplace transform of \(F.\) $$F(s)=\frac{e^{-4 s}(s+3)}{s^{2}-6 s+13}$$.

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}+5 y^{\prime}+4 y=20 \sin 2 t, \quad y(0)=1, \quad y^{\prime}(0)=-2\).

Determine a function \(f(t)\) that has the given Laplace transform \(F(s)\). $$F(s)=\frac{2}{s\left(s^{2}+16\right)}$$

Determine a function \(f(t)\) that has the given Laplace transform \(F(s)\). $$ F(s)=\frac{2 s+5}{s\left(s^{2}+4 s+20\right)} $$

Determine the Laplace transform of \(f\). $$f(t)=2 e^{3 t} \sin t+4 e^{-t} \cos 3 t$$.

Determine the inverse Laplace transform of \(F.\) $$F(s)=\frac{e^{-s}(2 s-1)}{s^{2}+4 s+5}$$.

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}-3 y^{\prime}+2 y=3 \cos t+\sin t, \quad y(0)=1, \quad y^{\prime}(0)=1\).

Sketch the given function and determine whether it is piecewise continuous on \([0, \infty)\). $$f(t)=\left\\{\begin{aligned} 3, & 0 \leq t \leq 1 \\ 0, & 1 \leq t < 3 \\ -1, & t \geq 3 \end{aligned}\right.$$

Express \(L^{-1}[F(s) G(s)]\) in terms of a convolution integral. $$F(s)=\frac{s+4}{s^{2}+8 s+25}, \quad G(s)=\frac{s e^{-\pi / 2}}{s^{2}+16}$$

Determine the Laplace transform of \(f\). $$f(t)=e^{2 t}\left(1-\sin ^{2} t\right)$$.

Determine the inverse Laplace transform of \(F.\) $$F(s)=\frac{2 e^{-2 s}}{(s-1)\left(s^{2}+1\right)}$$.

Determine a function \(f(t)\) that has the given Laplace transform \(F(s)\). $$F(s)=\frac{2 s+5}{s\left(s^{2}+4 s+20\right)}$$

Express \(L^{-1}[F(s) G(s)]\) in terms of a convolution integral. $$F(s)=\frac{1}{s-4}, \quad G(s) \text { arbitrary }$$

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}+4 y=9 \sin t, \quad y(0)=1, \quad y^{\prime}(0)=-1\).

Sketch the given function and determine whether it is piecewise continuous on \([0, \infty)\). $$f(t)=\left\\{\begin{array}{cc} t, & 0 \leq t \leq 1 \\ 1 / t^{2}, & t > 1 \end{array}\right.$$

Solve the given initial-value problem up to the evaluation of a convolution integral. $$y^{\prime \prime}+y=e^{-t}, \quad y(0)=0, \quad y^{\prime}(0)=1$$

Determine the Laplace transform of \(f\). $$f(t)=t^{2}\left(e^{t}-3\right)$$.

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}+y=6 \cos 2 t, \quad y(0)=0, \quad y^{\prime}(0)=2\).

Sketch the given function and determine whether it is piecewise continuous on \([0, \infty)\). $$f(t)=\left\\{\begin{array}{cc} 1, & 0 \leq t \leq 1 \\ 1 /(t-1), & t > 1 \end{array}\right.$$

Sketch \(f(t),\) express \(f(t)\) in terms of \(u_{a}(t),\) and determine \(L\\{f(t)\\}\). $$f(t)=\left\\{\begin{array}{cr} 2, & 0 \leq t < 1 \\ 3-t, & t \geq 1 \end{array}\right.$$

Determine the inverse Laplace transform of \(F.\) $$F(s)=\frac{50 e^{-3 s}}{(s+1)^{2}\left(s^{2}+4\right)}$$.

Sketch \(f(t),\) express \(f(t)\) in terms of \(u_{a}(t),\) and determine \(L\\{f(t)\\}\). $$f(t)=\left\\{\begin{array}{rr} 1, & 0 \leq t < \ln 2 \\ 2 e^{-t}, & t \geq \ln 2 \end{array}\right.$$

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

Determine the Laplace transform of \(f\). $$f(t)=e^{-2 t} \sin (t-\pi / 4)$$.

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}+9 y=7 \sin 4 t+14 \cos 4 t, \quad y(0)=1, \quad y^{\prime}(0)=2\).

Sketch the given function and determine whether it is piecewise continuous on \([0, \infty)\). $$f(t)=t, \quad 0 \leq t<1, \quad f(t+1)=f(t)$$

Solve the given initial-value problem up to the evaluation of a convolution integral. $$y^{\prime \prime}-2 y^{\prime}+10 y=\cos 2 t, \quad y(0)=0, \quad y^{\prime}(0)=1$$

Sketch \(f(t),\) express \(f(t)\) in terms of \(u_{a}(t),\) and determine \(L\\{f(t)\\}\). $$f(t)=\left\\{\begin{array}{cc} t, & 0 \leq t < 1 \\ 1 & 1 < t \leq 2 \\ 3-t, & 2 < t \leq 3 \\ 0, & t > 3 \end{array}\right.$$

Use the Laplace transform to find the general solution to \(y^{\prime \prime}-y=0\).

Sketch the given function and determine whether it is piecewise continuous on \([0, \infty)\). $$f(t)=n, \quad n \leq t < n+1, \quad n=0,1,2, \ldots$$

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

Solve the given initial-value problem up to the evaluation of a convolution integral. \(y^{\prime \prime}+16 y=f(t), \quad y(0)=\alpha, \quad y^{\prime}(0)=\beta,\) where \(\alpha\) and \(\beta\) are constants.

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

Let \(f \in E(0, \infty)\) and let \(a\) be a positive real number. Define the function \(f_{a}\) as follows $$ f_{a}(t)=\left\\{\begin{array}{cl} f(t-a), & \text { if } t \geq a \\ 0, & \text { if } 0 \leq t < a \end{array}\right. $$ Show that \(L\left\\{f_{a}(t)\right\\}=f(s+a)\)

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

Use the Laplace transform to solve the initial-value problem $$\begin{aligned}y^{\prime \prime}+\omega^{2} y &=A \sin \omega_{0} t+B \cos \omega_{0} t \\\y(0) &=y_{0}, \quad y^{\prime}(0)=y_{1}\end{aligned}$$ where \(A, B, \omega,\) and \(\omega_{0}\) are positive constants and \(\omega \neq \omega_{0}\).

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

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

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

Use the Convolution Theorem and the table of Laplace transforms to show that $$ \int_{0}^{x}(x-w)^{a} w^{b} d w=\frac{a ! b !}{(a+b+1) !} x^{a+b+1} $$ \(a > -1, b > -1, x > 0\)