91Ó°ÊÓ

Show that for \(n\) a nonnegative integer $$ L^{-1}\left\\{\frac{1}{(s+a)^{n+1}}\right\\}=\frac{t^{n} e^{-a t}}{n !} $$

Solve the problem by the Laplace transform method. Verify that your solution satisfies the differential equation and the initial conditions. $$ x^{\prime \prime}(t)-4 x^{\prime}(t)+4 x(t)=4 e^{2 t} ; x(0)=-1, x^{\prime}(0)=-4 $$

Solve the problem by the Laplace transform method. Verify that your solution satisfies the differential equation and the initial conditions. $$ x^{\prime \prime}(t)+x(t)=6 \sin 2 t ; x(0)=3, x^{\prime}(0)=1 $$

Express \(F(t)\) in terms of the \(\alpha\) function and find \(L\\{F(t)\\}\). $$ \begin{aligned} F(t) &=t^{2}, & 03 \end{aligned} $$

Show that for \(m>-1\) $$ L^{-1}\left\\{\frac{1}{(s+a)^{m+1}}\right\\}=\frac{t^{m} e^{-a t}}{\Gamma(m+1)} $$

Solve the following equation for \(F(t)\) with the condition that \(F(0)=4\) : \(F^{\prime}(t)=t+\int_{0}^{t} F(t-\beta) \cos \beta d \beta\)

Solve the problem by the Laplace transform method. Verify that your solution satisfies the differential equation and the initial conditions. $$ y^{\prime \prime}(t)-y(t)=4 \cos t ; y(0)=0, y^{\prime}(0)=1 $$

Express \(F(t)\) in terms of the \(\alpha\) function and find \(L\\{F(t)\\}\). $$ \begin{aligned} F(t) &=e^{-t}, & 02 \end{aligned} $$

Solve the problem by the Laplace transform method. Verify that your solution satisfies the differential equation and the initial conditions. $$ y^{\prime \prime}(t)-6 y^{\prime}(t)+9 y(t)=6 t^{2} e^{3 t} ; y(0)=y^{\prime}(0)=0 $$

Show that $$ L^{-1}\left\\{\frac{s}{(s+a)^{2}+b^{2}}\right\\}=\frac{1}{b} e^{-a t}(b \cos b t-a \sin b t) $$