Assume solutions of the equation
![\[ (x+1)y' + y^2 = 0 \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-698803c5b00a3c6b6982a6a525ae270b_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*} (x+1)y' + y^2 = 0 && \implies && y' \left( \frac{1}{y^2} \right) &= \frac{-1}{x+1} \\[9pt] && \implies && \int \frac{1}{y^2} \, dy &= -\int \frac{1}{x+1} \, dx \\[9pt] && \implies && -\frac{1}{y} &= -\log |x+1| + C \\[9pt] && \implies && y \big( \log |x+1| + C \big) &= 1. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-285e1fdd3a6df7741511b63e15f4d282_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