(a) Given the differential equation

\[\frac{d^{2} x}{d t^{2}}+7 \frac{d x}{d t}+5 x=8 u(t) \quad t \geq 0\]

Using MATLAB program, find

(i) $x(t)$ when all the initial conditions are zero

(ii) $x(t)$ when $x(0)=1$ and $\dot{x}(0)=3$.

(b) Given the differential equation

\[\frac{d^{2} x}{d t^{2}}+12 \frac{d x}{d t}+15 x=35 \quad t \geq 0\]

Using MATLAB program, find

(i) $x(t)$ when all the initial conditions are zero

(ii) $x(t)$ when $x(0)=0$ and $\dot{x}(0)=1$.

(iii) For the following differential equation, use MATLAB to find $x(t)$ when $x(t)$ when $x(0)=-1$ and $\dot{x}(0)=1$

\[\frac{d^{2} x}{d t^{2}}+8 \frac{d x}{d t}-4 x=18 u(t)\]

(c) For the following differential equation, use MATLAB to find $x(t)$ when $x(t)$ when $x(0)=-1$ and $\dot{x}(0)=1$

\[\frac{d^{2} x}{d t^{2}}+15 \frac{d x}{d t}+8 x=-9 u(t)\]

(d) For the following differential equation, use MATLAB to find $x(t)$ when $x(t)$ when $x(0)=-1$ and $\dot{x}(0)=1$

\[\frac{d^{2} x}{d t^{2}}+19 \frac{d x}{d t}+9 x=-3 u(t)\]