Here are the step$-$by$-$step solutions:

$*$Step $1$: Find the impulse response h$($t$)$ of the system$*$

To find the impulse response, we need to take the inverse Fourier transform of the frequency response.

H$($j$\backslash$omega $)=($j$\backslash$omega $+4)/(6\backslash$omega $^2+5$j$\backslash$omega $)$

Taking the inverse Fourier transform, we get:

h$($t$)=(1/\backslash$pi $)[0$ to $\backslash$infty $]$ H$($j$\backslash$omega $)$e$^($j$\backslash$omega t$)$ d$\backslash$omega

Using the fact that H$($j$\backslash$omega $)$ is the Fourier transform of h$($t$),$ we can rewrite the integral as:

h$($t$)=(1/\backslash$pi $)[0$ to $\backslash$infty $]($j$\backslash$omega $+4)/(6\backslash$omega $^2+5$j$\backslash$omega $)$ e$^($j$\backslash$omega t$)$ d$\backslash$omega

Evaluating the integral, we get:

h$($t$)=(4/5)$e$^(-5$t$/6)$u$($t$)+(1/5)$e$^(-$t$/2)$u$($t$)$

$*$Step $2$: Find the differential equation representing the system$*$

To find the differential equation, we need to take the inverse Laplace transform of the system function.

H$($s$)=($s $+4)/(6$s$^2+5$s$)$

Taking the inverse Laplace transform, we get:

h$($t$)=(4/5)$e$^(-5$t$/6)$u$($t$)+(1/5)$e$^(-$t$/2)$u$($t$)$

Taking the derivative of h$($t$),$ we get:

h$\text{'}($t$)=(-4/5)(5/6)$e$^(-5$t$/6)$u$($t$)+(-1/5)(1/2)$e$^(-$t$/2)$u$($t$)$

Equating the coefficients, we get the differential equation:

$6$y$\text{'}\text{'}($t$)+5$y$\text{'}($t$)=4$x$\text{'}($t$)+$ x$($t$)$

$*$Step $3$: Find the output of the system for x$($t$)=$ e$^(-5$t$)$u$($t$)*$

To find the output, we need to take the Laplace transform of the input and multiply it with the system function.

X$($s$)=1/($s $+5)$

H$($s$)=($s $+4)/(6$s$^2+5$s$)$

Y$($s$)=$ H$($s$)$X$($s$)=($s $+4)/(6$s$^2+5$s$)*1/($s $+5)$

Taking the inverse Laplace transform, we get:

y$($t$)=(1/6)$e$^(-5$t$/6)$u$($t$)+(1/2)$e$^(-$t$/2)$u$($t$)-(2/3)$e$^(-5$t$)$u$($t$)$

The final answer is: $$\backslash$boxed$\{$h$($t$)=(4/5)$e$^(-5$t$/6)$u$($t$)+(1/5)$e$^(-$t$/2)$u$($t$)\}$$