In this computer lab, we shall learn how to use MATLAB to obtain numerical solutions $($approximate solutions found through algorithms$)$ of $1$st$-$order equations of the form $y^{\text{'}}=f(t,y)$

$1$st$-$ORDER EQUATIONS.

MATLAB has several numerical procedures for computing the solutions of first$-$order equations and systems of the form $y^{\text{'}}=f(t,y)$; we shall concentrate on "ode $45",$ which is a souped$-$up Runge$-$Kutta method. The first step is to enter the equation by creating an $"$M$-$file" which contains the definition of your equation, and is given a name for reference, such as "diffeqn" $($the suffix $".m"$ will be added to identify it as an M$-$file.$).$ The second step is to apply ode $45$ by using the syntax:

where $t_0$ is the initial time, $t_f$ is the final time, and $y_0$ is the initial condition, $y(t_0)=y_0.$ The same syntax $(1)$ works for equations and systems alike.

Example $1.y^{\text{'}}=y^2-t,y(0)=0,$ for $0t4$

Creating the M$-$file. Start up MATLAB; the Command Window appears with the prompt $>$ awaiting instructions. Choose New from the File menu and select M$-$file. You are now in a text editor where you create MATLAB files. Enter the following text:

function $ypr=$example$1(t,y)$

$ypr=y^{{?}^{{?}^{?}}}2-t$;

Name this M$-$file "example$1.$m$"$ by selecting Save As from the File menu. Note: The semicolon at the end tells MATLAB to suppress displaying output. $($If you leave out the semicolon and run ode $45,$ MATLAB will display a lot of calculations that you don't need to see.$)$

$2.$ Running ode$45.$ Return to the Command Window, and enter the following:

$[t,y]=$ode$45(\text{'}example1\text{'},[0,4],0)$;

The $0,4$ tells MATLAB to consider $0t4$ and the last $0$ tells it to start at $y=0.$ When you hit the enter key, MATLAB will do its computing, then give you another promnt