Differential Equation with sketching graph I'm trying to solve the equation:
$$
y' = \frac{x\left(x^2+1\right)}{4y^3},\,\,y(0) = -\frac{1}{\sqrt{2}}\tag{1}
$$
a) Find explicit solution and determine the interval in which the solution is defined.
b) Sketch a rough graph of the solution.
I tried:
$$
\int 4y^3dy = \int x^3+x dx.\tag{2}
$$
Eq.(2) has a solution of the form,
$$
y^4 = \frac{1}{4}x^4+\frac{1}{2}x^2+C.
$$
What am I suppose to do next?

Note that the RHS is $\frac 14(x^2+1)^2$, (using the contour condition) so this is really $2y^2=x^2+1$. What does this look like to you?
