Estimate the following solutions using Euler's method with...

Estimate the following solutions using Euler's method with $n=5$ steps over the interval $t=[0,1] .$ If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler’s method. How accurate is Euler’s method? $$y^{\prime}=3 t-y, y(0)=1 . \quad \text { Exact solution } y=3 t+4 e^{-t}-3$$

Estimate the following solutions using Euler's method with $n=5$ steps over the interval $t=[0,1] .$ If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler’s method. How accurate is Euler’s method?
$$y^{\prime}=3 t-y, y(0)=1 . \quad \text { Exact solution } y=3 t+4 e^{-t}-3$$

Key Concepts

Euler's Method

Euler's Method is a numerical technique used to approximate solutions to ordinary differential equations (ODEs) with a given initial value. It works by using the derivative, which predicts the rate of change of the solution, to estimate the solution at successive small time steps along the interval of integration.

Initial Value Problems (IVPs)

An Initial Value Problem involves finding a function that satisfies a differential equation along with an initial condition that specifies the value of the function at a starting point. IVPs are a fundamental concept in differential equations, where methods like Euler’s provide approximate solutions when analytic techniques may be challenging or impossible.

Step Size and Interval Partitioning

In numerical methods such as Euler's method, the interval of integration is divided into small steps. The step size, determined by the number of steps or the width of each subinterval, directly affects the accuracy of the approximation; smaller step sizes typically result in more accurate estimates because they reduce the error per step.

Error Analysis and Accuracy

Error analysis in the context of numerical methods involves assessing how approximations deviate from the exact solution. In Euler's method, the local truncation error per step can accumulate across steps, affecting the overall accuracy. Comparing the numerical result with an exact solution, when it is known, is a crucial aspect of understanding the limitations and performance of numerical methods.

Transcript

00:01 For this problem, we're going to be using oilers method to approximate the y value that corresponds to an x of 1.

00:08 And so we're told that we want to use five intervals.

00:13 So that means that n is equal to 5.

00:17 And because our range goes from, or our domain goes from 0 to 1, that tells us that our h value, or our step size is going to be 1 divided by 5, which will be 0 .2.

00:30 So we can go ahead and use this to fill out our column for x of n values.

00:36 So we know that essentially each time we increase our n by 1, we're going to increase our x by 0 .2.

00:44 So that means that our x values will be 0 .2, 0 .4, 0 .6, 0 .8, and 1.

00:54 And now we're going to use oilers equation or oilers method in order to fill out the right side of our column.

01:02 And so we know that in order to approximate our y value, we're going to use this formula right here.

01:10 Our y sub n is equal to y sub, okay, so i'll write this as y sub 1 is equal to y sub n minus 1, which will be y sub 0, plus h, which is 0 .2, times f of, x -0 y -0 which is going to be three times x -0 which is zero minus y -s of zero so minus one and when we plug in all of our values here we get y sub 1 is equal to 1 plus 0 .2 times negative 1 and we're going to get y sub 1 is equal to 0 .8 so we'll fill that in right there and we can do the same thing to calculate y -sub 2 y 2 will be equal to y sub 1 plus 0 .2 times 3 times x of 1 minus y sub 1.

02:07 So when we fill in our values here, we get y sub 2 is equal to 0 .8 plus 0 .2 times 2 times 0 .2.

02:17 So i'll go ahead and write that.

02:19 Just write that out right there.

02:20 Oops...

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