Use Euler's method to approximate the indicated function value to 3 decimal places, using h=0.1

Use Euler's method to approximate the indicated function...

Key Concepts

Step Size

The step size, often denoted as h in numerical methods such as Euler's method, is the increment in the independent variable used to compute successive approximations of the solution. A smaller step size generally improves the accuracy of the approximation, while a larger step size may lead to larger errors.

Separable Differential Equation

A separable differential equation is one in which the variables can be separated on different sides of the equation. This structure allows the equation to be rewritten such that all terms involving one variable (and its differential) are on one side, and all terms involving the other variable are on the other side, facilitating integration and the derivation of an exact solution.

Comparison of Numerical and Exact Solutions

Comparing numerical approximations to exact analytical solutions is a key concept in numerical analysis. This comparison allows one to assess the error inherent in the numerical method and to understand the trade-offs between simplicity and precision. Such comparisons are fundamental to evaluating and verifying the performance of numerical methods like Euler's method.

Euler's Method

Euler's method is a numerical technique used to approximate solutions of ordinary differential equations with a given initial condition. It works by stepping forward a short distance (the step size) and using the derivative (from the differential equation) to estimate the function's change over that interval. This method is basic but broadly applicable, especially for problems where an analytical solution is difficult to obtain.

Initial Value Problem

An initial value problem (IVP) in the study of differential equations specifies the value of the unknown function at a particular point, usually given as y(x0) = y0. This specification, combined with the differential equation itself, determines a unique solution that can be approximated numerically by methods like Euler's method or found analytically, as applicable.

Transcript

0:00 Hi there.

00:01 In this problem, we are asked to use euler's method to find an approximate solution for a differential equation, and then we will check it by finding the actual solution.

00:12 So let's start with euler's method.

00:14 We'll need our function g, which is just dy, the x, and in this case, we are given that it is y times e to the x.

00:24 Okay.

00:26 We also have the initial condition, y of zero equals two.

00:31 We know that.

00:32 That our h value is 0 .1 and we are supposed to find an approximate value for y of 0 .4.

00:46 Alright, so let's start with the other's method.

00:49 To make it easier, let's make a little table for ourselves.

00:53 So at each step we will need the x value, the y value, we'll need to plug those into our g.

01:03 Whatever we obtain there we will multiply by h which in this case is point one and finally we will add that to our current y value to get the next y value and then just keep repeating until we're done so our initial condition y of 0 equals 2 that tells us that when x is 0 y is 2 we can actually fill in the rest of the x values here since h is 0 .1 will those keep adding that to a for x values and will be done when we arrive at 0 .4.

01:40 Okay, so now we can start on the top row.

01:44 If we plug in 0 for x and 2 for y into y e to the x, we will get 2 times e to the 0, which is 2.

01:55 We'll take that, multiply by 0 .1, gives us 0 .2, and finally we'll add that to our current y value of 2 to get 2 .2.

02:07 So that's our new y value.

02:09 We will copy that down into the next row and start all over again.

02:14 So if we plug in 0 .1 for x, 2 .2 for y, our g will evaluate to 2 .43414.

02:23 Make sure you get that.

02:25 And then times point 1, we'll get 0 .2431.

02:30 Add that to our current y value, we'll get 2 .4431.

02:36 All right, so that's our new y value.

02:40 And do the same thing again.

02:43 This time when we plug into g, make sure you're getting 2 .98040.