Section 1
Test exercise
Using the integral definition, find the Laplace transtorms for each of the following:(a) $f(x)=8$(b) $f(x)=e^{3 x}$(c) $f(x)=-4 e^{2 \cdots 3}$
Using the Table of Laplace transforms, find the inverse Laplace transforms of each of the following:(a) $F(s)=-\frac{5}{(s-2)^{2}}$(b) $F(s)=\frac{2 e^{3}}{s^{3}}$(c) $F(s)=\frac{3}{s^{2}+9}$(d) $F(s)=-\frac{2 s-5}{s^{2}+3}$
Given that the Laplace transform of $x e^{-k}$ is $F(s)=\frac{1}{(s+k)^{2}}$ derive the Laplace transform of $x^{2} e^{3 x}$ without using the integral definition.
Use the Laplace transform to solve each of the foltowing equations:(a) $f^{\prime}(x)+2 f(x)=x$ where $f(0)=0$(b) $f^{\prime}(x)-f(x)=e^{-x}$ where $f(0)=-1$(c) $f^{\prime \prime}(x)+4 f^{\prime}(x)+4 f(x)=e^{-2 x}$ where $f(0)=0$ and $f^{\prime}(0)=0$(d) $4 f^{\circ}(x)-9 f(x)=-18$ where $f(0)=0$ and $f^{\prime}(0)=0$