Chapter 4

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

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

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

In Problems, use the Laplace transform to solve the given initial-value problem. $$ y^{\prime \prime}-6 y^{\prime}+9 y=t, \quad y(0)=0, \quad y^{\prime}(0)=1 $$

Use the Laplace transform to solve the given initial-value problem. $$ y^{\prime \prime}-6 y^{\prime}+9 y=t, \quad y(0)=0, \quad y^{\prime}(0)=1 $$

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

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

In Problems, use the Laplace transform to solve the given initial-value problem. $$ y^{\prime \prime}-6 y^{\prime}+13 y=0, \quad y(0)=0, \quad y^{\prime}(0)=-3 $$

Use the Laplace transform to solve the given initial-value problem. $$ y^{\prime \prime}-6 y^{\prime}+13 y=0, \quad y(0)=0, \quad y^{\prime}(0)=-3 $$

In Problems, use the Laplace transform to solve the given initial-value problem. $$ 2 y^{\prime \prime}+20 y^{\prime}+51 y=0, \quad y(0)=2, \quad y^{\prime}(0)=0 $$

Use the Laplace transform to solve the given initial-value problem. $$ 2 y^{\prime \prime}+20 y^{\prime}+51 y=0, \quad y(0)=2, \quad y^{\prime}(0)=0 $$

In Problems, use the Laplace transform to solve the given initial-value problem. $$ y^{\prime \prime}-y^{\prime}=e^{t} \cos t, \quad y(0)=0, \quad y^{\prime}(0)=0 $$

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

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

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

Use the Laplace transform to solve the given initial-value problem. $$ \frac{d y}{d t}-y=1, \quad y(0)=0 $$

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

A 4-lb weight stretches a spring \(2 \mathrm{ft}\). The weight is released from rest 18 in above the equilibrium position, and the resulting motion takes place in a medium offering a damping force numerically equal to \(\frac{7}{8}\) times the instantaneous velocity. Use the Laplace transform to find the equation of motion \(x(t)\).

Use the Laplace transform to solve the given initial-value problem. $$ y^{\prime}+6 y=e^{4 t}, \quad y(0)=2 $$

Use the Laplace transform to solve the given equation. $$ y^{\prime \prime}-2 y^{\prime}+y=e^{t}, \quad y(0)=0, \quad y^{\prime}(0)=5 $$

Recall that the differential equation for the instantaneous charge \(q(t)\) on the capacitor in an \(L R C\) -series circuit is $$ L \frac{d^{2} q}{d t^{2}}+R \frac{d q}{d t}+\frac{1}{C} q=E(t) $$ See Section 3.8. Use the Laplace transform to find \(q(t)\) when \(L=1 \mathrm{~h}, R=20 \Omega, C=0.005 \mathrm{f}, E(t)=150 \mathrm{~V}, t>0, q(0)=0\) and \(i(0)=0\). What is the current \(i(t) ?\)

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

Use the Laplace transform to solve the given equation. $$ y^{\prime \prime}-8 y^{\prime}+20 y=t e^{t}, \quad y(0)=0, \quad y^{\prime}(0)=0 $$

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

Use the Laplace transform to solve the given equation. $$ y^{\prime \prime}+6 y^{\prime}+5 y=t-t 9(t-2), \quad y(0)=1, y^{\prime}(0)=0 $$

Use the Laplace transform to find the charge \(q(t)\) in an \(R C\) -series when \(q(0)=0\) and \(E(t)=E_{0} e^{-k t}, k>0 .\) Consider two cases: \(k \neq 1 / R C\) and \(k=1 / R C .\)

Use the Laplace transform to solve the given initial-value problem. $$ y^{\prime \prime}-4 y^{\prime}=6 e^{3 t}-3 e^{-t}, \quad y(0)=1, y^{\prime}(0)=-1 $$

Use the Laplace transform to solve the given equation. $$ y^{\prime}-5 y=f(t), \text { where } f(t)=\left\\{\begin{array}{lr} t^{2}, & 0 \leq t<1 \\ 0, & t \geq 1, \end{array} \quad y(0)=1\right. $$

Use the Laplace transform to solve the given integral equation or integrodifferential equation. $$ f(t)+\int_{0}^{t}(t-\tau) f(\tau) d \tau=t $$

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

Find \(\mathscr{L}\\{f(t)\\}\) by first using an appropriate trigonometric identity. \(f(t) \quad \sin 2 t \cos 2 t\)

Use the Laplace transform to solve the given equation. $$ y^{\prime}(t)=\cos t+\int_{0}^{t} y(\tau) \cos (t-\tau) d \tau, \quad y(0)=1 $$

Use the Laplace transform to solve the given integral equation or integrodifferential equation. $$ f(t)=2 t-4 \int_{0}^{t} \sin \tau f(t-\tau) d \tau $$

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}\left\\{e^{2-t} q(t-2)\right\\} $$

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

Find \(\mathscr{L}\\{f(t)\\}\) by first using an appropriate trigonometric identity. \(f(t) \quad \cos ^{2} t\)

Use the Laplace transform to solve the given integral equation or integrodifferential equation. $$ f(t)=t e^{t}+\int_{0}^{t} \tau f(t-\tau) d \tau $$

In Problems, find either \(F(s)\) or \(f(t)\), as indicated. $$ \mathscr{L}\left\\{t^{\boldsymbol{q}} U(t-2)\right\\} $$

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

Find \(\mathscr{L}\\{f(t)\\}\) by first using an appropriate trigonometric identity. \(f(t) \quad \sin (4 t+5)\)

Use the Laplace transform to solve each system. $$ \begin{aligned} &x^{\prime}+y=t \\ &4 x+y^{\prime}=0 \\ &x(0)=1, y(0)=2 \end{aligned} $$

Use the Laplace transform to solve the given integral equation or integrodifferential equation. $$ f(t)+2 \int_{0}^{t} f(\tau) \cos (t-\tau) d \tau=4 e^{-t}+\sin t $$

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

Find \(\mathscr{L}\\{f(t)\\}\) by first using an appropriate trigonometric identity. \(f(t) \quad 10 \cos (t-\pi / 6)\)

Use the Laplace transform to solve each system. $$ \begin{aligned} &x^{\prime \prime}+y^{\prime \prime}=e^{2 t} \\ &2 x^{\prime}+y^{\prime \prime}=-e^{2 t} \\ &x(0)=0, \quad y(0)=0 \\ &x^{\prime}(0)=0, \quad y^{\prime}(0)=0 \end{aligned} $$

Use the Laplace transform to solve the given integral equation or integrodifferential equation. $$ f(t)+\int_{0}^{t} f(\tau) d \tau=1 $$

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

One definition of the gamma function \(\Gamma(\alpha)\) is given by the improper integral $$ \Gamma(\alpha) \quad \int_{0}^{\infty} t^{\alpha-1} e^{-t} d t, \alpha>0 $$ Use this definition to show that \(\Gamma(\alpha+1) \quad \alpha \Gamma(\alpha)\).

The current \(i(t)\) in an \(R C\)-series circuit can be determined from the integral equation $$ R i+\frac{1}{C} \int_{0}^{t} i(\tau) d \tau=E(t) $$ where \(E(t)\) is the impressed voltage. Determine \(i(t)\) when \(R=10 \Omega, C=0.5 \mathrm{f}\), and \(E(t)=2\left(t^{2}+t\right)\).

Use the Laplace transform to solve the given integral equation or integrodifferential equation. $$ f(t)=\cos t+\int_{0}^{t} e^{-\tau} f(t-\tau) d \tau $$