Use a numerical solver and Euler's method to obtain a four-decimal approximation of the indicate

Use a numerical solver and Euler's method to obtain a...

Key Concepts

Initial Value Problems

An initial value problem involves a differential equation along with an initial condition, which specifies the value of the unknown function at a starting point. This concept is essential as it guarantees a unique solution under appropriate conditions and focuses on how the solution evolves from the initial state.

Numerical Methods for Differential Equations

Numerical methods provide algorithmic approaches to approximate solutions of differential equations when analytical solutions are hard to obtain. They are critical in practical applications where precise models are built by computing approximate values rather than exact answers.

Euler's Method

Euler's Method is one of the simplest numerical techniques used to solve ordinary differential equations by using the slope at a given point to estimate the function's value at the next point. This step-by-step approach is fundamental in understanding more advanced numerical methods.

Step Size

The step size, typically denoted by h, represents the interval between successive points in the numerical approximation. It directly affects the accuracy of the approximation; smaller step sizes tend to yield better accuracy but increase the computational effort, while larger step sizes may reduce accuracy by introducing a larger error per step.

Iteration

Iteration refers to the process of applying the numerical method repeatedly over the interval of interest. In Euler's Method, each step uses the previous result to calculate the next value, highlighting the cumulative nature of error and the importance of careful step size selection.

Transcript

00:01 So in this question we are given a differential equation that is y prime is equal to x minus y whole square.

00:12 And we have an initial condition of y of zero is equal to 0 .5.

00:18 And we have to find out the values till y gives a value of 0 .5.

00:28 Fine.

00:29 So now what we are supposed to do via, we are supposed to find out the four decimal approximation.

00:42 Fine.

00:44 Using the numerical solver along with the eilers method.

01:05 Fine.

01:07 So let's start with a solution.

01:12 Now we know that according to the euler's method.

01:19 Find according to the euler's method.

01:23 We have an equation that is y n sub plus 1 is equal to y sub n plus h times f of x n y n.

01:38 Find where this x n is equal to x0 plus n times h and x and this n is equal to 0 1 2 and so on fine so from this above equation from this equation find what we can do over here is start to solve this question now from this method we have ym plus 1 is equal to y n plus h now what is the function over here that has been given to us is h times x n negative y n whole square right so what we are going to do over here as we are required in the question to first solve for a is equal to point one so we are going to do that so when h is equal to point one we have x not that is equal to zero and y not that is equal to 0 .5, n that is equal to 0.

03:07 So by plugging in these values, we find that 1 is equal 0 .5 to 5 to 5.

03:15 5 .5.

03:17 And when n is equal to 1, we get that x1 is equal to x0 plus 1 times 0 .1.

03:30 Plug in the values and we find that x1 is equal to 0 .1 right so similarly using this method we will find out the values when we'll have y2 that is approximately equal to 0 .5 431 and similarly the value for x2 that would be so in n would be equal to 2 so we'll have a value for y2 that would be this one and x2 would be equal to point two fine so similarly we'll solve it and therefore we can construct a table over here so in this table we have two columns right the first one is for x n and the other one is for yn so so so when x1 holds a value of 0 .0 .0 .0 .0.

04:55 Then we have a value for 0 .1, that is 0 .5 -250...

Please give Ace some feedback