Homework $3$: Submit hand calculation of finding the complete solution, code file for plotting the response, and the screenshot of the plot.

A system is given as follows:

$x+2x^2+3x=sint+(t-)$

$x(0)=0,x^{}(0)=1$

The complete solution after $t>$ is as follows:

Use initial conditions: $x(0)=0$ and $x^{}(0)=1$ to find unknown constants A and $\frac{O}{?}$ in the complete solution and and plot the response.

$2.$ The force function of a spring mass damper system is given:

$m{x}^{}(t)+c{x}^{}(t)+kx(t)=F(t)$

$F(t)=\{\begin{array}{cc}1,& 0t\end{array}$

$-1,t2$

The period of the square wave is T$.$ Determine the Fourier series representation, calculate and plot the total response of the spring$-$mass$-$damper system with $m=100kg,=0\mathrm{.}1$ and $k=1000\frac{N}{m}$ to the signal.

$3.$ Repeat problem $2$ where the force function is given by a saw tooth curve as follows:

The saw tooth curve of period $T$ is

$F(t)=\frac{1}{2}t,0t2$