Separable Initial Value Problem

[hardyaa1]

The IVP is (1 + x^2)(dy/dx) = x y^3 and y(0)=1.

After separating, On the right side I end up having to do a substitution with u = 1+x^2,
after that it gets kinda messy to get the explicit answer, I just wanted to check and see if I'm doing something wrong.

Thanks,
-A

 

[galactus]

\(\displaystyle (1+x^{2})\frac{dy}{dx}=xy^{3}\)

Upon separating variables, we get:

\(\displaystyle \int \frac{1}{y^{3}}dy=\int\frac{x}{1+x^{2}}dx\)

\(\displaystyle \frac{-1}{2y^{2}}=\frac{ln(x^{2}+1)}{2}+C\)

To find C, sub in y=1 and x=0 and solve for C.