$1$ Van der Pol Equation

The second order nonlinear autonomous differential equation

$\frac{d^{2}y}{d{t}^2}+(y^2-1)\frac{dy}{dt}+y=0,>0$

is called the van der Pol equation. It describes many physical systems collectively called van der Pol

oscillators. The equation models a non$-$conservative system in which energy is added to and subtracted

from the system, resulting in a periodic motion called a limit cycle. The parameter $$ is a positive scalar

indicating the non$-$linearity and the strength of the damping.

Please solve the van der Pol equation for the following initial conditions:

$y(0)=2,dydt(0)=0$

and $=0\mathrm{.}01,0\mathrm{.}1,1,5,10,$ respectively.

Instructions

i$.$ Write down in paper the ODE given in Eq$.1$ as an equivalent first$-$order system of ODEs and

specify the corresponding initial conditions.

ii$.$ Write down in paper the forward Euler method to the first$-$order system of ODEs obtained in $($i$),$

derive the discrete equations for evaluating y and dy$/$dt at t $=$ ti $=$ i h$,$ where h is the time step

size and i is an integer number.

iii. Wite down the $4$th$-$order Runge$-$Kutta method to the first$-$order system of ODEs obtained in $($b$),$

derive the discrete equations for evaluating y and dy$/$dt at t $=$ ti $=$ i h$,$ where h is the time step

size and i is an integer number. iv$.$ Implement the forward Euler method in Python or Matlab or any other language to solve the firstorder system of ODEs obtained in $($i$)$ with the discrete equations obtained in $($b$)$ over t

using h $=0\mathrm{.}01,0\mathrm{.}005,0\mathrm{.}001,$ respectively. Specifically, for each $$ and h$,$ plot three figures:

a$.$ y as function of t$,$ in the span of time t

b$.$ dy$/$dt as function of t$,$ in the span of time t

c$.$ dy$/$dt as function of y$.$

v$.$ Implement the $4$th$-$order Runge$-$Kutta method in Python or Matlab or any other language to solve

the first$-$order system of ODEs obtained in $($i$)$ with the discrete equations obtained in $($b$)$ over

t using h $=0\mathrm{.}01,0\mathrm{.}005,0\mathrm{.}001,$ respectively. Specifically, for each $$ and h$,$ plot three

figures:

a$.$ y as function of t$,$ in the span of time t

b$.$ dy$/$dt as function of t$,$ in the span of time t

c$.$ dy$/$dt as function of y$.$