Chapter 10

Determine \(f(t-a)\) for the given function \(f\) and the given constant \(a\). $$f(t)=e^{-t} \sin 2 t, \quad a=\pi / 6$$.

Make a sketch of the given function on \([0, \infty)\) and express it in terms of the unit step function. $$f(t)=\left\\{\begin{aligned}3, & 0 \leq t<1, \\\\-1, & t \geq 1\end{aligned}\right.$$

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

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

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

$$\text { Prove that } f *(g+h)=f * g+f * h$$

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

Determine \(f(t-a)\) for the given function \(f\) and the given constant \(a\). $$f(t)=\frac{t}{t^{2}+4}, \quad a=1$$.

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

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

Use \((10.10 .1)\) to determine \(L[f]\). $$f(t)=\left\\{\begin{aligned} t+1, & 0 \leq t < 3 \\ t^{2}-1, & t > 3 \end{aligned}\right.$$

Make a sketch of the given function on \([0, \infty)\) and express it in terms of the unit step function. $$f(t)=\left\\{\begin{array}{lr}t^{2}, & 0 \leq t<1 \\\1, & t \geq 1\end{array}\right.$$

Use (10.1.1) to determine \(L[f]\). $$f(t)=\left\\{\begin{array}{cc} t^{2}, & 0 \leq t \leq 1 \\ 1, & t > 1 \end{array}\right.$$

Solve the given initial-value problem. $$y^{\prime \prime}+6 y^{\prime}+13 y=\delta(t-\pi / 4), \quad y(0)=5, \quad y^{\prime}(0)=5$$

Determine the Laplace transform of the given function \(f.\) $$f(t)=e^{-2(t-1)} \sin 3(t-1) u_{1}(t)$$.

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

Make a sketch of the given function on \([0, \infty)\) and express it in terms of the unit step function. $$f(t)=\left\\{\begin{aligned}2, & 0 \leq t<2 \\\1, & 2 \leq t<4 \\\\-1, & t \geq 4 \end{aligned}\right.$$

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}-3 y^{\prime}+2 y=4, \quad y(0)=0, \quad y^{\prime}(0)=1\).

Determine the inverse Laplace transform of the given function. $$F(s)=\frac{5}{s+3}.$$

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

Determine \(L[f * g]\) $$f(t)=t, \quad g(t)=\sin t$$

Use properties of the Laplace transform and the table of Laplace transforms to determine \(L[f]\). $$f(t)=5 \cos 2 t-7 e^{-t}-3 t^{6}$$

Solve the given initial-value problem. $$\begin{aligned} &y^{\prime \prime}+9 y=15 \sin 2 t+\delta(t-\pi / 6), \quad y(0)=0\\\ &y^{\prime}(0)=0 \end{aligned}$$

Determine \(L[f * g]\) $$f(t)=e^{2 t}, \quad g(t)=1$$

Determine \(f(t-a)\) for the given function \(f\) and the given constant \(a\). $$f(t)=e^{-t}(\sin 2 t+\cos 2 t), \quad a=\pi / 4$$.

Determine the Laplace transform of the given function \(f.\) \(f(t)=e^{a(t-c)} \cos b(t-c) u_{c}(t),\) where \(a, b,\) and \(c\) are positive constants.

Make a sketch of the given function on \([0, \infty)\) and express it in terms of the unit step function. $$f(t)=\left\\{\begin{array}{cc}2, & 0 \leq t<1 \\\2 e^{(t-1)}, & t>1\end{array}\right.$$

Determine the inverse Laplace transform of the given function. $$F(s)=\frac{4}{s^{3}}.$$

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}-y^{\prime}-12 y=36, \quad y(0)=0, \quad y^{\prime}(0)=12\).

Determine \(L[f * g]\) $$f(t)=\sin t, \quad g(t)=\cos 2 t$$

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

Make a sketch of the given function on \([0, \infty)\) and express it in terms of the unit step function. $$f(t)=\left\\{\begin{array}{cc}t, & 0 \leq t<3 \\\6-t, & 3 \leq t<6 \\\0, & t \geq 6 \end{array}\right.$$

Use properties of the Laplace transform and the table of Laplace transforms to determine \(L[f]\). $$f(t)=e^{-5 t} / \sqrt{t}$$

Solve the given initial-value problem. $$\begin{aligned} &y^{\prime \prime}+16 y=4 \cos 3 t+\delta(t-\pi / 3), \quad y(0)=0\\\ &y^{\prime}(0)=0 \end{aligned}$$

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}+y^{\prime}-2 y=10 e^{-t}, \quad y(0)=0, \quad y^{\prime}(0)=1\).

Determine the inverse Laplace transform of the given function. $$F(s)=\frac{2 s}{s^{2}+9}.$$

Use properties of the Laplace transform and the table of Laplace transforms to determine \(L[f]\). $$f(t)=e^{3 t} \cos 5 t-e^{-t} \sin 2 t$$

Solve the given initial-value problem. $$\begin{aligned} &y^{\prime \prime}+2 y^{\prime}+5 y=4 \sin t+\delta(t-\pi / 6), \quad y(0)=0\\\ &y^{\prime}(0)=1 \end{aligned}$$

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

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

Make a sketch of the given function on \([0, \infty)\) and express it in terms of the unit step function. $$f(t)=\left\\{\begin{array}{cc}0, & 0 \leq t<2, \\\3-t, & 2 \leq t<4 ,\\\\-1, & t \geq 4, \end{array}\right.$$

Use the linearity of \(L\) and the formulas derived in this section to determine \(L[f]\). $$f(t)=2 \sin 3 t+4 t^{3}$$

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}-3 y^{\prime}+2 y=4 e^{3 t}, \quad y(0)=0, \quad y^{\prime}(0)=0\).

Determine the inverse Laplace transform of the given function. $$F(s)=\frac{2 s+1}{s^{2}+16}.$$

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

Determine \(f(t)\). $$f(t-2)=(t-2) e^{3(t-2)}$$.

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

Use the Laplace transform to solve the given initial-value problem. \(y^{\prime \prime}-2 y^{\prime}=30 e^{-3 t}, \quad y(0)=1, \quad y^{\prime}(0)=0\).

Determine the inverse Laplace transform of the given function. $$F(s)=\frac{s+6}{s^{2}+1}.$$

Use properties of the Laplace transform and the table of Laplace transforms to determine \(L[f]\). $$f(t)=6 t^{4} e^{-2 t}-2 t e^{t+1}+\sqrt{10 t}$$