How could I find a solution to the differential equation
$$x\frac{dy}{dx}=y^2-y$$
that passes through the point $y=(1/2,1/2)$
and write it in terms of $y$ ?
What I have attempted: I rewrote the differential equation into the form
$$\frac{1}{y^2-y}dy=\frac{1}{x}dx$$
and then integrated it
$$\int \frac{dy}{y^2-y}=\frac{1}{x}dx,$$
getting $$\ln\frac{y-1}{y}=\ln(x).$$
$$e^{ln((y-1)/y)}=e^{ln(x)}+c$$
$$\frac{y-1}{y}=x+c$$
$$y-1=xy+cy$$
$$y-1=y(x+c)$$
But now how can we make pass it through $(1/2,1/2)$ in terms of $y$ ?
Best Answer
Put $(x=\frac{1}{2},y=\frac{1}{2})$ in the equation you have obtained and find the value of the integration constant $c$. That will serve your problem. As much as I see it, the value of $c$ will come out to be $-\frac{3}{2}$.