The equation is separable. That is to say, we're given a differential equation of the form
[tex]\dfrac{\mathrm dy}{\mathrm dx}=f(x,y)[/tex]
but it happens that we can write [tex]f(x,y)=g(x)h(y)[/tex], a product of functions of their own independent variables. When this is the case, we can split up the differentials:
[tex]\dfrac{\mathrm dy}{\mathrm dx}=g(x)h(y)\implies\dfrac{\mathrm dy}{h(y)}=g(x)\,\mathrm dx[/tex]
Then we integrate both sides and attempt to solve for [tex]y(x)[/tex].
For this ODE, we can write
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{x^2}{y^2}\implies y^2\,\mathrm dy=x^2\,\mathrm dx[/tex]
[tex]\implies\displaystyle\int y^2\,\mathrm dy=\int x^2\,\mathrm dx[/tex]
[tex]\implies\dfrac{y^3}3=\dfrac{x^3}3+C[/tex]
[tex]\implies y^3=x^3+C[/tex]
[tex]\implies y=(x^3+C)^{1/3}=\sqrt[3]{x^3+C}[/tex]
Other methods are available, but I think they tend to be outside the scope of AP Calc curriculum.
