Numerical solver for y’=3xy^3

Question

For this equation is the Taylor Series the best approach to wstimate numerically? I am also using Euler’s to compare. 

 

Trying to find y(1)

given y(x) is a solution to the equation and knowing y(0)=1

Answer 1

I suggest integration.

 

y’ = 3xy³

 

y'/y³ = 3x

 

y^-3dy = 3xdx

 

y^-2/2 = 3x²/2 + c

 

y(0) = 1 ⇒

 

(1)^-2/2 = 3(0)²/2 + c

 

1/2 = c

 

y^-2/2 = 3x²/2 + 1/2 = (1/2)(3x² + 1)

 

y = √[4/(3x² + 1)]

 

Now plug in x = 1 to get y(1).