Use (a) Euler’s method and (b) Improved Euler\'s Method to find approximate valu

Question

Use (a) Euler’s method and (b) Improved Euler's Method to find approximate values of the solution of the given initial value problem at the points xi = x0 +  ih, where x0 is the point where the initial condition is imposed and i = 1, 2, 3.

The answers I'm getting using Euler and then Improved Euler are not even remotely close so I must be doing something wrong with the pi and sin or something?

but for Euler use y1=y0+h[f(x0,y0)]

and for Improved Euler use: y1=y0 + h/2[ f(x0,y0)+ f(x1, y1)] where you would use the original euler formula to solve for y1 to find f(x1,y1) to plug in and find the actual y1 value.

Please show steps and explain! Thanks!

Explanation / Answer

C) Given, y’+x^2y=sin(xy)

So, we can write, y’=sin(xy)-x^2y

Now, let f(x, y)=sin(xy)-x^2y

By Euler’s method we have,

y_i+1=y_n+h*f(x_i, y_i)

and, x_i=x_0 + i*h

Given y(1)=, and h=0.2, so we get, x_0 =1

First iteration,

x_1=x_0 +1*h=1+1*0.2=1.2

y_1=y_0 +h*f(x_0, y_0)

= +0.2*f(1, )

=2.51327

Second Iteration: i=2,

x_2=x_0 +2*h=1+2*0.2=1.4

y_2=y_1+h*f(x_1, y_1)

= 2.51327 +0.2*f(1.2, 2.51327)

=1.81452

Third iteration: i=3

x_3=x_0 +3*h=1+3*0.2=1.6

y_3=y_2+h*f(x_2, y_2)

= 1.81452 +0.2*f(1.4, 1.81452)

=1.21637

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