Question

What is the answer to this?
Find the approximate solution of the initial value problem presented below
in terms of a Taylor polynomial of degree 4.

y' + x^2-y^2 = 0, y(0) = 1

Answer 1

Solve for y'(x) and you'll see that y'(x)=-x^2+y(x)^2
We know from the problem:
y(0)=1
y'(0)=0+1^2=1

I'll help you get started on the next one even, y''(x)=-2x+2y(x)y'(x)

y''(0)=?
y'''(0)=?

You just need to approximate it to 4 and then put it into a taylor polynomial. Don't forget the whole point of a taylor polynomial is to approximate them so that they have the same first, second, ... nth derivative at x=0.

Answer 2

I hope that helped... Now that I reread the question I'm not sure if it will or not lol.