(Solved):   a) Using definition of Laplace transform to find Laplace transform of \[ f(t)=\left\{\begi ...

 

a) Using definition of Laplace transform to find Laplace transform of \[ f(t)=\left\{\begin{array}{ll} 0, & 0 \leq t<\frac{2 \pi}{3}, \\ t-\frac{2 \pi}{3}, & t \geq \frac{2 \pi}{3} . \end{array}\right. \] (5 marks) b) Consider the piecewise function \[ g(t)=\left\{\begin{array}{ll} 2, & 0 \leq t<4, \\ e^{t-2}, & t \geq 4 . \end{array}\right. \] Write \( g(t) \) in terms of unit step function. Hence, find \( \mathcal{L}\{g(t)\} \). (5 marks) c) Use convolution theorem to find \[ \mathcal{L}^{-1}\left\{\left(\frac{s}{s^{2}+16}\right)^{2}\right\} . \] (5 marks) QUESTION 4 (15 MARKS) a) Use Laplace transform to solve the initial value problem \[ y^{\prime \prime}+y^{\prime}+y=e^{3 t} \delta(t-\pi), \quad y(0)=0, y^{\prime}(0)=0 . \] (5 marks) b) Use Laplace transform to solve the system of differential equations \[ \begin{array}{l} \frac{d x}{d t}=y-H \\ \frac{d y}{d t}=9 x, \end{array} \] given that \( x(0)=1 \) and \( y(0)=-1 \).