91Ó°ÊÓ

Find the convolution \(f(t) * g(t)\). $$ f(t)=t^{2}, g(t)=\cos t $$

Find the convolution \(f(t) * g(t)\). $$ f(t)=e^{a t}, g(t)=e^{b t} $$

Apply the convolution theorem to find the inverse Laplace transforms of the functions. $$ F(s)=\frac{s^{2}}{\left(s^{2}+4\right)^{2}} $$

Apply Duhamel's principle to write an integral formula for the solution of each initial value problem in Problems. $$ x^{n}+4 x^{\prime}+8 x=f(t) ; x(0)=x^{\prime}(0)=0 $$

Apply the convolution theorem to find the inverse Laplace transforms of the functions. $$ F(s)=\frac{1}{s\left(s^{2}+4 s+5\right)} $$

Use partial fractions to find the inverse Laplace transforms of the functions. $$ F(s)=\frac{5 s-4}{s^{3}-s^{2}-2 s} $$

Use Laplace transforms to solve the initial value problems in Problem. $$ x^{\prime \prime}+6 x^{\prime}+25 x=0 ; x(0)=2, x^{\prime}(0)=3 $$

In Problems, the values of the elements of an RLC circuit are given, Solve the initial value problem $$ L \frac{d i}{d t}+R i+\frac{1}{C} \int_{0}^{t} i(\tau) d \tau=e(t) ; \quad i(0)=0 $$ with the given impressed voltage e \((t) .\) $$ \begin{aligned} &L=1, R=150, C=2 \times 10^{-4} ; e(t)=100 t \text { if } 0 \leqq t<1 ; \\ &e(t)=0 \text { if } t \geqq 1 \end{aligned} $$