91Ó°ÊÓ

Solve the problem by the Laplace transform method. Verify that your solution satisfies the differential equation and the initial conditions. $$ y^{\prime \prime}+a^{2} y=0 ; y(0)=0, y^{\prime}(0)=a $$

In exercises 1 through 10 obtain \(L^{-1}\\{f(s)\\}\) from the given \(f(s)\). $$ \frac{s}{s^{2}+4 s+4} $$

Find an inverse transform of the given \(f(s)\). $$ \frac{5 s-2}{s^{2}(s+2)(s-1)} $$

Solve the problem by the Laplace transform method. Verify that your solution satisfies the differential equation and the initial conditions. $$ y^{\prime \prime}-3 y^{\prime}+2 y=e^{3 t} ; y(0)=y^{\prime}(0)=0 $$

Find an inverse transform of the given \(f(s)\) using the convolution theorem. $$ \frac{1}{\left(s^{2}+1\right)^{2}} $$

In exercises 1 through 10 obtain \(L^{-1}\\{f(s)\\}\) from the given \(f(s)\). $$ \frac{2 s-3}{s^{2}-4 s+8} $$

Solve the given equation. If sufficient time is available, verify your solution. $$ F(t)=1+2 \int_{0}^{t} F(t-\beta) \cos \beta d \beta $$

Express \(F(t)\) in terms of the \(\alpha\) function and find \(L\\{F(t)\\}\). $$ \begin{aligned} F(t) &=3, & 01 \end{aligned} $$

Solve the problem by the Laplace transform method. Verify that your solution satisfies the differential equation and the initial conditions. $$ y^{\prime \prime}+y=e^{-t} ; y(0)=y^{\prime}(0)=0 $$

In exercises 1 through 10 obtain \(L^{-1}\\{f(s)\\}\) from the given \(f(s)\). $$ \frac{3 s+1}{s^{2}+6 s+13} $$