(i) Use the definition of the Laplace transform of $f(t)$ :

$$L\{f(t)\}=F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$

to show that, for $f(t)=t^{n}$,

$$L\{f(t)\}=F(s)=\frac{n !}{s^{n+1}}, \quad L\left\{e^{a t} f(t)\right\}=F(s-a)=\frac{n !}{(s-a)^{n+1}}$$

(ii) Use contour integration to find the inverse Laplace transform of

$$F(s)=\frac{1}{s^{2}(s+1)^{2}}$$

(iii) Verify the result in (ii) by using the results in (i) and the convolution theorem.

(iv) Use Laplace transforms to solve the differential equation

$$f^{(i v)}(t)+2 f^{\prime \prime \prime}(t)+f^{\prime \prime}(t)=0$$

subject to the initial conditions

$$f(0)=f^{\prime}(0)=f^{\prime \prime}(0)=0, \quad f^{\prime \prime \prime}(0)=1$$