MATLAB CODE NEEDED

Euler's method for solving Ordinary Differential Equation $($ODE$)$ y$^()($t$)=($dy$($t$))/($dt$)=$f$($y$($t$))$ over the interval f$\_(0,$t$),$f$\_($f$)$ with initial condition y$($f$\_(0))=$y$0$ is based on the approximation formula

q$,$y$($t$)=($dy$($t$))/($dt$)$~~$($y$($t$+$dt$)-$y$($t$))/($dt$)=$f$($y$($t$))$

which results in the recursive formula

q$,\{($y$($t$\_(0)+($k$+1)*$dt$)=$y$($t$\_(0)+$k$*$dt$)+$dt$*$f$($y$($t$\_(0)+$k$*$dt$)))$:$\}$

y$($f$\_(0))=$y$\_(0)$

or equivalently

$\{($y$\_($k$+1)=$y$\_($k$)+$dt$*$f$($y$\_($k$)))$:$\}$

y$\_(0)$ IC

where y$\_($k$)=$y$($t$\_(0)+$k$-$dt$)$ for all k$>=0$

Remember the solution of an ODE is a function y:$|$ta$,$tf$|->$R$,$ over the entire domain $.$ The sequence $\{($y$\_($k$)|\_($k$)0)$:$\}$ is a piecewise constant function that approximates the actual solution, and depends on the $($discretization$)$ time step dt$.$

Consider the following ordinary differential equation

$($dy$)/(($d$)$t$)=2$t$+3$ with initial condition y$=1$ at t$=$ to$.$

Write a Matlab function ode$\_($e$)$uler that

Computes the y function values using Euler's method from t$=$t$\_(0)$ to t$=$t$\_($f$)$ with the step size of dt$.$

Takes as input parameters the initial time t$\_(0),$ the final time t$\_($f$),$ the initial value of the function y$\_(0).$ and the step size dt$.$

Function $$

Code to call your function $(3)$

l

$3$ y y $=1$; $\%$xax y value at t$=0($initial value$)$

dt $=0.$e$01$; $\%$step size

$5$

$6$