91Ó°ÊÓ

Prove that L_2yand L_2sby convolution integral

A transformation of a function into the function that is useful especially in reducing the solution of an ordinary linear differential equation with constant coefficients to the solution of a polynomial equation.

The inverse Laplace transform of a function F(s)is the piecewise-continuous and exponentially-restricted real function f(t)

Laplace transform of L₂₇and L₂₈

$$\begin{gathered} L_{28}:L^{-1}[e^{pu}.G(p\left)\right]=g(t-a,)t>a>0=0,t<a \\ {L}_{27}:L^{-1}\left(e^{pu}\right)=\mathrm{δ}(t-a);aâ‰\yen 0 \\ {L}^{-1}[e^{-pa}-G(p\left)\right]=L^{-1}\left(G\right(p).e^{-pu}) \\ =L^{-1}\left[G\right(p).H(P\left)\right] \end{gathered}$$

Solve further

$$\begin{gathered} =g*h \\ ={∫}_0^{1}g\left(\mathrm{τ}\right).h(t-\mathrm{τ})d\mathrm{τ} \\ ={∫}_0^{1}g(t-\mathrm{τ}).h\left(t\right)d\mathrm{τ} \end{gathered}$$

$$\begin{gathered} H\left(p\right)=e^{-pa} \\ L\left(h\right(r\left)\right)=e^{-pa}h\left(t\right) \\ =L^{-1}\left(e^{-pa}\right) \\ h\left(t\right)=\mathrm{δ}(t-a) \end{gathered}$$

$h\left(t\right)=\mathrm{δ}(\mathrm{Ï„}-a),aâ‰\yen 0$

$$\begin{gathered} L^{-1}(e^{-pa}-G(p\left)\right)={∫}_0^{1}g(t-\mathrm{Ï„}).\mathrm{δ}(r-a) \\ {L}^{-1}({\mathrm{Î\mu }}^{-pa}-G(p\left)\right)=g(t-a),0<a<t \\ =0,t<\mathrm{an}L<1(e^{-pa}-G(p\left)\right) \\ =g(t-a),t>a>0=0,t<a \end{gathered}$$