IN MATLAB

a$.)$ Consider the modified problem

$\frac{d^{2}y}{d{t}^2}+6y^2\frac{dy}{dt}+7y=2sint,$ with $,y(0)=-1,\frac{dy}{dt}(0)=0.$

The ODE $(7)$ is very similar to $(4)$ except for the $y^2$ term in the left$-$hand side. Because of the factor $y^2$ the ODE $(7)$ is nonlinear, while $(4)$ is linear. There is however very little to change in the implementation of $(4)$ to solve $(7).$ In fact, the only thing that needs to be modified is the ODE definition.

Modify the function defining the ODE in LAB$04$ex$1.$m$.$ Call the revised file LABO$4$ex$2.$m$.$ The new function M$-$file should reproduce the pictures in Fig $8.$

Include in your report the changes you made to LAB$04$ex$1.$m to obtain LABO$4$ex$2.$m$.$

Figure $8$: Time series $y=y(t)$ and $v=v(t)=y^{\text{'}}(t)($left$),$ and phase plot $v=y^{\text{'}}$ vs$.y$ for $(7).$

$($b$)$ Compare the output of Figs $7$ and $8.$ Describe the changes in the behavior of the solution in the short term.

$($c$)$ Compare the long term behavior of both problems $(4)$ and $(7),$ in particular the amplitude of oscillations.

$($d$)$ Modify LABO$4$ex$2.$m so that it solves $(7)$ using Euler's method with $N=400$ in the interval $0t50($use the file euler.m from LAB $3$ to implement Euler's method; do not delete the lines that implement ode$45).$ Let $te,Ye$ be the output of euler, and note that $Ye$ is a matrix with two columns from which the Euler's approximation to $y(t)$ must be extracted. Plot the approximation to the solution $y(t)$ computed by ode$45($in black$)$ and the approximation computed by euler $($in red$)$ in the same window $($you do not need to plot $v(t)$ nor the phase plot$).$ Are the solutions identical? Comment. What happens if we increase the value of $N?$