Solve this differential equation with the initial condition y(0) = 0

Question

dy/dx = y^2 * x^(2/3)

I am having trouble solving for the constant "C" because I always get an undefined result. How would I solve this?

Answer 1

If $y(x) = 0$ for all $x$, then $\frac{dy}{dx} =\, ?$ and $y(x)^2 \,x^{2/3} = \,?$.

Answer 2

Well first we take the derivative of your solution, which is zero since ${{d}\over{dx}}y(x)={d\over{dx}}0=0$. Plugging this back into your equation, we get $${{dy}\over{dx}}=0=y^2*x^{2/3}.$$ Since $y=0$, $$0=0^2*x^{2/3}=0.$$ Thus we have proven that your equation is a solution. This also solves the initial condition since $y(x)=0$ for any choice of x