Solving Higher$-$Order Differential Equations

Consider the following equation: $\frac{d^{2}y}{d{t}^2}+\frac{dy}{dt}=y+1$

It can be reformulated into a system of equations by introducing a new variable $z.$ Let $z=\frac{dy}{dt}$ It is easy to see that $\frac{dz}{dt}=\frac{d^{2}y}{d{t}^2}.$ Substituting into original equation we get $\frac{dz}{dt}+\frac{dy}{dt}=y+1$ which is a first order differential equation.

Effectively we've replaced $\frac{d^{2}y}{d{t}^2}+\frac{dy}{dt}=y+1$ with the following two equations:

$\frac{dy}{dt}=z$ and $\frac{dz}{dt}=y+t-\frac{dy}{dt}$

Create an m $-$file function to use in the ode$45$ solver. The function should have two inputs, which are typically called $t$ and $y.$ The variable $t$ is the independent variable, and $y$ is an array of dependent variables.

The range of time is defined as $-1$ to $+1.$

Initial conditions are defined as $y=0$ and $z=0$

You should have a legend in the southwest corner

You should label your axis

You should have a title on the graph

You can choose any colors or line style

$($In matlab$)$