Problem Four (12 Marks) Use Runge Kutta method of order four...

Problem Four (12 Marks) Use Runge Kutta method of order four to approximate the solution of the initial value problem y' + 2y = xe^3x, 0 ? x ? 1, y(0) = 0, with h = 0.5 Hint: Compute y(0.5) and y(1)

Transcript

00:01 So we have given here the differential equation that is y dash plus 2y is equal to x e to the power 3x and we have given here some condition that is x is less than equal to 2 1 and greater than equal to 2 0 and h is equal to we have here 0 .5.

00:29 So by using the runge -kutta method, runge -kutta method we have here yn plus 1 is equal to yn is equal to yn plus 1 by 6 multiply with k1 plus 2k2 plus 2k3 and plus we need to write here k4.

01:09 So from here and let's this is our equation number first.

01:13 So where k1 is equal to we have here that is h function of xn comma yn where we need to write k2 is equal to hf function of xn plus h by 2 comma yn plus k1 by 2.

01:47 Now we need to write our k3 value that is equal to hf of xn plus h by 2 comma yn plus k2 by 2 and now we need to write here the value of k4 that is equal to hf into xn plus h comma yn plus k3.

02:29 Now we need to find first that is we have to write here our function value that is y dash is equal to x into e to the power 3x minus 2y.

02:45 So from here we need to write our function f xn comma yn is equal to xn e to the power 3xn minus 2yn.

03:03 Now from to above equation k1 is equal to then value of h is 0 .5 we have given here and x0 we need to put here n is equal to 0.

03:18 Put n is equal to 0.

03:22 So from here that is x0 e to the power 3x0 minus 2 into y0.

03:37 So after simplify this we will get from here that is 0 .5 and x0 value is 0 e to the power 3 into 0 minus y0 value is again 0.

03:52 So our k1 value is we have here a 0.

03:56 Now we need to find here our k2 value so according to formula we need to write here 0 .5 f and that is x0 plus 0 .5 by 2 comma y0 plus 0 because k1 upon 2 become 0 here.

04:24 So after simplify this we will get from here that is 0 .5 and it becomes from here that is we will get our value here 0 .25 multiply with e to the power 3 0 .25 minus 2 into 0.

04:59 So we will get from here our answer that is 0 .264625.

05:07 Now we need to find our k3 value that is equal to then 0 .5 f of we need to write x0 plus 0 .5 upon 2 comma y0 plus 0 .2646 by 2.

05:40 So we will get from here after simplify that is 0 .5 f because x0 value become 0...

Question

Problem Four (12 Marks) Use Runge Kutta method of order four to approximate the solution of the initial value problem y' + 2y = xe^3x, 0 ? x ? 1, y(0) = 0, with h = 0.5 Hint: Compute y(0.5) and y(1)

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