Assume solutions of the equation
![\[ y' = \frac{x^3}{y^2} \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-b282452fd38d3ef8d7dfbef79c2df683_l3.png "Rendered by QuickLaTeX.com")
exist and find an implicit formula satisfied by these solutions.
This is a separable first-order equation. We compute,
![\begin{align*} y' = \frac{x^3}{y^2} && \implies && y^2 y' &= x^3 \\ && \implies && \int y^2 \, dy &= \int x^3 \, dx \\[9pt] && \implies && \frac{1}{3}y^3 &= \frac{1}{4} x^4 + C \\[9pt] && \implies && y^3 &= \frac{3}{4} x^4 + C. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-b243617e9d1e74758eb101d1cef8659e_l3.png "Rendered by QuickLaTeX.com")
Assume solutions of the equation
exist and find an implicit formula satisfied by these solutions.
This is a separable first-order equation. We compute,