We studied the ODEs for the Pendulum and the simple spring. A similar system is called the

Van der Pol oscillator given by

$y^{}-(1-y^2)y^{}+y=0$

It's used for a variety of models, but most famously for the electrical potential of a firing

Neuron.

$($a$)(10$ points$)$ Write this second order ODE and as a first order system of ODEs.

$($b$)(20$ points$)$ Code up the $2$nd order Runge$-$Kutta Method represented by the tableau in

problem $1$ with $b_2=\frac{3}{4}$ and solve the system derived above with the initial conditions

$y^{}(0)=0,y(0)=0$ and $=1,$ for $t=0$ to $t=10$ with $1000$ time points, i$.$e$.t=0\mathrm{.}01.$

Submit your code and have it produce a plot of your solution with the $y(t)$ on the $x-$axis

and $y^{}(t)$ on the $y-$axis as functions of time $t.$