Compute the true solution to the problem Y

To solve the given problem using Euler's method numerically, we need to approximate the solution to the differential equation $Y^{\prime }(t)=-e^{-t}Y(t)$ with initial condition $Y(0)=1$. We will use different step sizes $h$ of 0.2, 0.1, and 0.05.
Euler's method is an iterative numerical method that approximates the solution of a differential equation by taking small steps and using the derivative at each step.
Let's begin by calculating the approximate values of $Y(t)$ using Euler's method for each step size:
For $h=0.2$:
We start with the initial condition: $Y(0)=1$. Using the formula for Euler's method, we can iteratively update the value of $Y$ at each step.
For the first step:
$t_0=0$ and $Y_0=1$ (initial condition).
$Y_1=Y_0+h·Y^{\prime }(t_0)$
Substituting the given differential equation, we have:
$Y_1=1+0.2·(-e^{-0}·1)=1-0.2=0.8$
For the second step:
$t_1=0.2$ and $Y_1=0.8$.
$Y_2=Y_1+h·Y^{\prime }(t_1)$
Substituting the differential equation:
$Y_2=0.8+0.2·(-e^{-0.2}·0.8)\approx 0.8-0.0378\approx 0.7622$
Similarly, we can continue this process to find $Y_3,Y_4,\dots$ until we reach the desired value of $t$. Repeat this process for $h=0.1$ and $h=0.05$.
Once we have obtained the approximate values of $Y(t)$ for each step size, we can calculate the error and relative error using the true solution $Y(t)$.
To perform these calculations, it is recommended to use programming tools such as MATLAB, as it allows for efficient implementation of numerical methods. You can use MATLAB to write a code that iteratively solves the equation using Euler's method, calculates the true solution $Y(t)$, and then computes the error and relative error for each step size.