Chapter 10

Solve the given initial-value problem. $$y^{\prime}+y=\delta(t-5), \quad y(0)=3$$

Determine \(f * g\) $$f(t)=t, \quad g(t)=1$$

Determine the Laplace transform of the given function \(f.\) $$f(t)=u_{2}(t)-u_{3}(t)$$.

Determine \(f(t-a)\) for the given function \(f\) and the given constant \(a\). $$f(t)=2 t, \quad a=1$$.

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime}-2 y=6 e^{5 t}, y(0)=3\).

Show that the given function is of exponential order. $$f(t)=e^{2 t}.$$

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

Determine \(f * g\) $$f(t)=6 t^{2}, \quad g(t)=5 t^{3}$$

Determine the Laplace transform of the given function \(f.\) $$f(t)=(t-1) u_{1}(t)$$.

Determine \(f(t-a)\) for the given function \(f\) and the given constant \(a\). $$f(t)=e^{-2 t}, \quad a=-1$$.

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime}+y=8 e^{3 t}, \quad y(0)=2\).

Show that the given function is of exponential order. $$f(t)=\cos 2 t.$$

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

Determine \(f * g\) $$f(t)=\cos t, \quad g(t)=t$$

Determine \(f(t-a)\) for the given function \(f\) and the given constant \(a\). $$f(t)=1, \quad a=3$$.

Determine the Laplace transform of the given function. $$f(t)=\cos t, \quad 0 \leq t<\pi, \quad f(t+\pi)=f(t)$$

Use \((10.10 .1)\) to determine \(L[f]\). $$f(t)=4 t^{2}$$

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

Show that the given function is of exponential order. $$f(t)=t e^{-2 t}.$$

Solve the given initial-value problem. $$y^{\prime}-5 y=2 e^{-t}+\delta(t-3), \quad y(0)=0$$

Determine \(f * g\) $$f(t)=e^{t}, \quad g(t)=t$$

Determine \(f(t-a)\) for the given function \(f\) and the given constant \(a\). $$f(t)=t^{2}-2 t, \quad a=-2$$.

Make a sketch of the given function on the interval \([0, \infty)\). $$f(t)=t\left(1-u_{1}(t)\right)$$.

Determine the Laplace transform of the given function \(f.\) $$f(t)=\sin (t-\pi / 4) u_{\pi / 4}(t)$$.

Use (10.1.1) to determine \(L[f]\). \(f(t)=\sin b t,\) where \(b\) is constant.

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

Show that the given function is of exponential order. $$f(t)=e^{3 t} \sin 4 t.$$

Use \((10.10 .1)\) to determine \(L[f]\). $$f(t)=7 t e^{-t}$$

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

Determine \(f(t-a)\) for the given function \(f\) and the given constant \(a\). $$f(t)=e^{3 t}, a=2$$.

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

Determine \(f * g\) $$f(t)=t^{2}, \quad g(t)=e^{t}$$

Determine the Laplace transform of the given function \(f.\) $$f(t)=\cos t u_{\pi}(t)$$.

Make a sketch of the given function on the interval \([0, \infty)\). $$f(t)=u_{1}(t)+u_{2}(t)+u_{3}(t)+u_{4}(t)$$.

Show that the given function is of exponential order. \(f(t)=t^{n} e^{a t},\) where \(a\) and \(n\) are positive integers.

Solve the given initial-value problem. $$y^{\prime \prime}-4 y=\delta(t-3), \quad y(0)=0, \quad y^{\prime}(0)=1$$

Determine the Laplace transform of the given function. $$f(t)=\left\\{\begin{array}{ll}2 t / \pi, & 0 \leq t<\pi / 2 \\\\\sin t, & \pi / 2 \leq t<\pi\end{array}\right.$$ where \(f(t+\pi)=f(t)\).

Determine \(f * g\) $$f(t)=e^{t}, \quad g(t)=e^{t} \sin t$$

Determine \(f(t-a)\) for the given function \(f\) and the given constant \(a\). $$f(t)=e^{2 t} \cos t, \quad a=\pi$$.

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

Show that the given function is of exponential order. Show that if \(f\) and \(g\) are in \(E(0, \infty),\) then so are \(f+g\) and \(c f\) for any scalar \(c .\)

Solve the given initial-value problem. $$y^{\prime \prime}+2 y^{\prime}+5 y=\delta(t-\pi / 2), \quad y(0)=0, \quad y^{\prime}(0)=2$$

Determine \(f(t-a)\) for the given function \(f\) and the given constant \(a\). $$f(t)=t e^{2 t}, \quad a=-1$$.

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

Determine the Laplace transform of the given function. $$f(t)=\left\\{\begin{array}{rl}1, & 0 \leq t<1 \\\\-1, & 1 \leq t \leq 2,\end{array}, \text { where } f(t+2)=f(t)\right.$$

Make a sketch of the given function on the interval \([0, \infty)\). $$f(t)=u_{1}(t)-u_{2}(t)+u_{3}(t)-\cdots$$ $$=\sum_{i=1}^{\infty}(-1)^{i+1} u_{i}(t)$$

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

Determine the Laplace transform of the given function \(f.\) $$f(t)=(t-1)^{2} u_{2}(t)$$.

Use \((10.10 .1)\) to determine \(L[f]\). $$f(t)=\left\\{\begin{array}{lr} 2, & 0 \leq t \leq 1 \\ t, & t > 1 \end{array}\right.$$

Solve the given initial-value problem. $$y^{\prime \prime}-4 y^{\prime}+13 y=\delta(t-\pi / 4), \quad y(0)=3, \quad y^{\prime}(0)=0$$