Euler's Method is a method for approximating solutions to differential equations numerically.

The process is as follows. Pick a time$-$step $t,$ an initial value yo$,$ and an initial time $t_0.$ If

we are trying to solve the differential equation

$\frac{dy}{dt}=f(t,y),y(t_0)=y_0$

we do this by starting with our approximate solution at $t_0$ and approximating the value at

$t_1=t_0+t$ using a straight line with slope $f(t_0,y_0).$ That is$,$ we set

$y_1=y_0+(t)f(t_0,y_0)$

and this should approximate the value of the solution at $t_0+t.$ We can iterate this process,

setting

$y_2=y_1+(t)f(t_1,y_1)$

and, in general

$y_{n+1}=y_n+(t)f(t_n,y_n)$

which should approximate the value of the solution at $t_0+nt.$ There is a function

$[t,y]=$eulerMethod$(f,dt,Tf,t0,y0)$

written near the bottom of the MATLAB format file. There are a few lines missing from

the code that are annotated with comments. Fill in those lines with the code that nexds to

be there for the method to work properly. You are filling in the time$-$stepping part of the

process, which was described above, Keep in mind bow MATLAB arrays work. The lines are

as follows

ind is the current index of the array. This will step from $1$ to the number of iterations

needed. It is the $n$ in the equations above for Euler's Method.

$m$ is the slope of the tangent line$/$approximate line that we will use for this step. It is

computed as $f(t,y)$ for the current values of $t$ and $y.$

The $t$ array stores all of the $t$ values in sequence. The line starting with $t(ind+1)$ will

compute the next value of $t$ based on the previous value $t($ind$)$ and the timestep dt$.$

The $y$ array stores all of the $y$ values in sequence. The line starting with $y($ind$+1)$ will

compute the next value of $y$ hosed on the previous value $y($ind$),$ the timetep $dt,$ and

the slope $n.$

Now, consider the initial value problem

$\frac{dy}{dt}=e^t-t,y(0)=1$

which has solution

$y(t)=e^t-\frac{t^2}{2}.$

Use the eulerMethod function that you just completed to estimate this solution at $T_f=2$

using the $l$ value of $0\mathrm{.}2.$ You should also compute the error in this numerical approximation

by taking the absolute value of the difference between the numerical approximation at $2$ and

the value of the actual solution $y(t)$ at $t=2.$ You should display the array of solution values

$($y$)$ as well as this error. Notes on using this function:

The function gives two outputs, $t$ and $y$ in that order. This means that if you want to

access the $y$ values, you will need to ask for two outputs from the eulerMethod function,

say of the form $t,y-$ eulerMethod $(..),$ and then the output will be stored in $y.$

To get the last value of an array, you can use y $($end$).$ In the case of Enler's method,

this will give the approximate value at $t=2.$ This value can then be compared to $y(2)$

when written as a function in Matlab. Make sure you take the absolute value of the

difference, as the error should be positive.

Think about the output that you get from this method. How many values should there

be in each array? What should the list of $t$ values look like? Check to make sure that

your outputs match these if the results do not look right.