Use the Euler method with d x=1 / 3 to estimate y(2) if y^'= x siny and y(0)=1 . What is the exa

Use the Euler method with d x=1 / 3 to estimate y(2) if y^'=...

Key Concepts

Euler Method

Euler's Method is a straightforward numerical technique for approximating solutions to ordinary differential equations. It works by advancing a solution over small increments using the derivative's value (the slope) at the previous point, thereby creating a sequence of approximate values that estimate the solution curve.

Step Size

The step size (denoted as dx or h) is the fixed interval between points in numerical methods like Euler's Method. It determines how finely the interval of interest is divided, with smaller step sizes typically improving the accuracy of the approximation at the expense of requiring more computational steps.

Initial Value Problem

An initial value problem involves finding a function that satisfies an ordinary differential equation along with a given initial condition. The initial condition uniquely determines the solution curve of the differential equation and is essential for applying numerical methods.

Separation of Variables

Separation of Variables is an analytical method used to solve certain differential equations by expressing the equation such that all terms involving the dependent variable are on one side and all terms involving the independent variable are on the other. This technique allows for the integration and determination of an exact solution when the equation is separable.

Transcript

00:01 So this question tells us to use the euler method with dx equals 1 3 to estimate y of 2, and we're told that y prime is equal to x sine of y, and y of 0 is equal to 1.

00:13 And then we're asked for the second part of the question, what is the exact value of y of 2? so there's really two parts of this question.

00:19 We're first using the euler method to estimate y of 2, and then we're going to find the exact value of y of 2.

00:25 So first, to start with what we're given.

00:26 We're given that y prime is equal to x sine of y.

00:33 We're also given the initial value that y at zero is equal to one, and that d x is equal to one -third.

00:45 So for the first part of the question, we want to estimate y of two using oilish method.

00:50 So the first thing when you remember is what exactly is oilish method.

00:53 So oil's method tells us that y sub n is equal to y n minus one, plus the five.

01:02 Function evaluated at x n minus 1 y n minus 1 times dx and then it also tells us that xn that's just going to equal to xn minus 1 plus d x so for this first part of the question i am going to first find all of my x ends so first off starting at x not here this just comes from our initial value and that's the first you tells me that x not is equal to 0 that y not is equal to 1 so of x not is equal to 0 and then from this equation we know that we're just going to add dx each time so add dx to 0 i just get one third and so on and then i want to stop at 1 before 2 klutzolorelish method tells us to do since we're doing like left end points so we have all of our x -ends now we can plug those into our equation for y so first y1 we know it's going to equal to y not plus the function evaluated x not y not times d x so y not you're given is equal to one so i'll do one plus i'm going to have my function here but i'm placing x with x not and y with y with y not and when i do that i get zero times sign of y not which again is one times dx which is one -third and so when i multiply add those up i just end up with one and then i'm just going to keep on going so now y -1 is equal to 1 plus x1 which is 1 3rd times sign of y1 which is 1 times 1 3rd and so this is going to end up equaling 1 .09 350 so now my y2 is equal to this number so i plug that in for here plus x2 which is two thirds sign of that same number times dx and get 1 .29989 and keep on going the same process and i'm plugging in y3 the number i just found x3 which is 1 and when you put that in your calculator you should get 1 .61125 and keep on going so now now my x4 is 4 thirds.

04:43 And when you calculate that out, you should get 2 .0553.

04:49 And then for the last one, i get putting in y5, x5 is 5 thirds.

05:07 And when you calculate that out, you should get 2 .54694.

05:13 And so this is the estimate of y of 2 using oilers method.

05:18 And then now for the second part, access to find the exact value of that.

05:22 So do that, we're going to solve the initial value problem.

05:26 So again, we're given that d, y, d, x times sine of y, and that y of zero is equal to 1.

05:35 So my first step to solve this, i'm going to go ahead and put my dy and dx on opposite sides, and put my sign of y with my dy, x with dx, and when i do that, i get 1 over sine of y, d, y, is equal to x d x.

05:54 Well, 1 over sine of y is the same thing as co -secant of y...

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