Problem $3$A $--$ Group A: ONLY the students with even email IDs $($e$.$g$.,$ sn$84$ or ab$450).$

Use Odeint package to solve the following equation $[15$ marks$]$

In Section $9$ of the course we learned how to integrate Ordinary Differential Equations. Use the examples given in Sec$09\_$ODEs to work through this problem.

Use odeint to solve the following third order ODE on a linear grid of $1000$ points from $x=0$ to $x=3$ :

$\frac{d^{3}y}{d{x}^3}+\frac{d^{2}y}{d{x}^2}+\frac{1}{t_s}\frac{dy}{dx}-Gy+\frac{3}{2}ysin(2y)=0$

here $y(x)$ is an unknown function, $t_s$ and $G$ are constant parameters. The initial conditions at point $x=0$ are: $y(0)=0,d\frac{y}{d}x(0)=1,$ and $d^2\frac{y}{d}{x}^2(0)=0.$

Solve this equation numerically using parameter values $G=1$ and $t_s=1.$ Plot functions $y(x)$ and $d\frac{y}{d}x$ on the same plot making it as informative and professionally

looking as possible. Find the maximum value of $y(x)$ on the plot, which we shall call $y_{\text{m}\text{a}\text{x}},$ and print it inside the plot window.

Now let us see how $y_{\text{m}\text{a}\text{x}}$ changes as the parameter $t_s$ changes. Create an array of $50t_s$ values covering the range from $t_s=0\mathrm{.}1$ to $t_s=10.$ For each value of $t_s$ in the

array repeat the calculation above and find the corresponding value of $y_{\text{m}\text{a}\text{x}}.$ You should now have $50$ values of $t_s$ and $50$ corresponding values of $y_{\text{m}\text{a}\text{x}}.$ Plot $y_{\text{m}\text{a}\text{x}}$ as a

function of $t_s$ in another plot window. Using the worked example: exoplanet in Section $08,$ fit the $y_{\text{m}\text{a}\text{x}}$ vs $t_s$ relationship with a polynomial of $5$th order, and display

it on the same plot.