Find the solution to $\frac{dy}{dx}=\left(\frac{-y}{x}\right)^{\frac{1}{3}}$ that goes through the point $(8,1)$.

Question

Find the solution to $\frac{dy}{dx}=\left(\frac{-y}{x}\right)^{\frac{1}{3}}$ that goes through the point $(8,1)$.

The answer is $x^\frac{2}{3} + y^\frac{2}{3}=5$ but I do not understand how to get there so any help is appreciated thanks!

Answer 1

We have $$ \frac{\mathsf dy}{\mathsf dx} = \left(\frac{-y}x\right)^{\frac13} = (-1)^{\frac13}\left(\frac yx\right)^{\frac13} = - \left(\frac yx\right)^{\frac13} \implies y^{-\frac13}\ \mathsf dy = -\left(\frac 1x\right)^{\frac13}\ \mathsf dx $$ and so integrating yields $$ \int y^{-\frac13}\ \mathsf dy = \int-\left(\frac 1x\right)^{\frac13}\ \mathsf dx \implies \frac 32 y^{\frac23} = -\frac32 x^{\frac 23} + C, $$ or $y^{\frac 23} + x^{\frac 23}= C$. The condition that $y(8)=1$ gives $$ 8^{\frac23} + 1^{\frac23} = C \implies 4+1 = C \implies C = 5, $$ and hence the solution is $$ x^{\frac23}+y^{\frac23} = 5. $$