Solve only question 3 interval01with a step size of t02 3...

solve only question 3 interval[0,1]with a step size of t=0.2. 3. problem by hand. Use forward Euler's method to approximate the solution on the time Now consider the IVP y=sint2y+y+y,y0=1.We cannot easily solve this Repeat problem 1 with a step size of t =0.1. Which step size gives better accuracy? the arithmetic. Compare your approximate value for y(1 with the true value of y1. on the interval [0.1] with a step size of t =0.2. You can use a calculator to help with 1.Use (forward Euler's method to approximate the solution of the IVP y=-3y,0=1

Transcript

00:01 In this question we have been given an initial value problem which is defined as y -daz equals 1 plus y over t where t is between 1 and 2 and the initial condition y at 1 equals to 2.

00:17 We need to use second order tailor polynomial method to approximate the solution okay so first of all let's see how we are going to do this and h value we have been given equals to h is basically the step size which is given as 1 over 3 which is approximately 0 .3 3 3.

00:39 Okay, so we need to find out what is y of 2.

00:45 So first of all, we need to find y at x equals to 2.

00:51 Y at t equals to 2 basically.

00:54 So what i will do first of all, i will replace this.

00:57 T with x so what i will get then i can say my x not is given to be one y not equals to two from this initial condition correct h value is 0 .3333 and we take xn value to be equals to 2 correct so let's say this is our equation number 1 so differentiate equation number 1 let's see what we are getting we get y double dash equals to now the derivative of this will be x y dash minus one minus y over x square okay this is what i get further we can simplify or further we can find out again the derivative of this and this comes out to be equals to x cube y double minus 2x y 2x square y dash plus 2x plus 2x y correct all divided with x raise to 4 and again we need the 4th derivative okay so the fourth derivative i will get as the following minus 3x raise to 6 y double dash plus x raise to 7 y triple dash plus 6x x raise to 5 y double dash plus 6x raise to 5 y dash minus 6 x raised to 4 minus 6 x raised to 4 times y all divided with x raised to 8 correct so now just have to substitute the values in these three okay so substitute the values which have been given okay we get the following y -daz to be equals to y y not of dash so y not of dash will be 1 plus y not over x not so it becomes three similarly why not double dash it comes out to be zero and why not triple dash it is also zero and why not at the fourth derivative okay so this is also coming out to be zero now what we can do so this is what we are getting okay so now substitute the above values in the taylor cs our values in the taylor series...

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