91Ó°ÊÓ

Use partial fractions to find the inverse Laplace transforms of the functions. \(F(s)=\frac{5 s-6}{s^{2}-3 s}\)

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

Use partial fractions to find the inverse Laplace transforms of the functions. \(F(s)=\frac{5-2 s}{s^{2}+7 s+10}\)

Find the Laplace transforms of the functions given in Problems 11 through \(22 .\) $$ f(t)=\sin t \text { if } 0 \leqq t \leqq 2 \pi ; f(t)=0 \text { if } t>2 \pi $$

This problem deals with a mass \(m\), initially at rest at the origin, that receives an impulse \(p\) at time \(t=0 .\) (a) Find the solution \(x_{\epsilon}(t)\) of the problem $$ m x^{\prime \prime}=p d_{0, \epsilon}(t) ; \quad x(0)=x^{\prime}(0)=0 . $$ (b) Show that \(\lim _{\epsilon \rightarrow 0} x_{\epsilon}(t)\) agrees with the solution of the problem $$ m x^{\prime \prime}=p \delta(t) ; \quad x(0)=x^{\prime}(0)=0 . $$ (c) Show that \(m v=p\) for \(t>0\) ( \(v=d x / d t\) ).

Use Laplace transforms to solve the initial value problems $$ x^{\prime}+2 y^{\prime}+x=0, x^{\prime}-y^{\prime}+y=0 ; x(0)=0, y(0)=1 $$

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

Find the Laplace transforms of the functions given in Problems 11 through \(22 .\) $$ f(t)=\cos \pi t \text { if } 0 \leqq t \leqq 2 ; f(t)=0 \text { if } t>2 $$

Verify that \(u^{\prime}(t-a)=\delta(t-a)\) by solving the problem $$ x^{\prime}=\delta(t-a) ; \quad x(0)=0 $$ to obtain \(x(t)=u(t-a)\).

Use Laplace transforms to solve the initial value problems $$ \begin{aligned} &x^{\prime \prime}+2 x+4 y=0, y^{\prime \prime}+x+2 y=0 ; x(0)=y(0)=0, \\ &x^{\prime}(0)=y^{\prime}(0)=-1 \end{aligned} $$