Assume solutions of the equation
![\[ y' = (y-1)(y-2) \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-a1d8495165118fb38c57c1db69cfa120_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' = (y-1)(y-2) && \implies && \frac{y'}{(y-1)(y-2)} &= 1 \\[9pt] && \implies && \int \frac{1}{(y-1)(y-2) \, dy} &= \int \, dx \\[9pt] && \implies && \log |y-2| - \log |y-1| &= x + C \\ && \implies && \frac{y-2}{y-1} &= Ce^x \\ && \implies && y-2 &= C(y-1)e^x. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-169c98846214b7bf47eedefb872b61c7_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