Consider the vector-field $\vec{F}=(x^2,y^2-y)$. Find an equation for the field-line that passes through the point $(1,\frac{1}{2})$.
This is what I have so far-
Let $\vec{r}(t)$ be the field line that passes through the point $(1,\frac{1}{2})$. Then
$$\frac{dx}{x^2}=\frac{dy}{y^2-y}$$
which after integrating both sides gives
$$-\frac{1}{x}=\ln|y-1|-\ln|y|+C_{1}.\quad\quad\quad\quad(1)$$
and this is where I get stuck. If I plug the point $(1,\frac{1}{2})$ in $(1)$ I get $\ln 0$ which is obviously wrong. If I let $C_{1}=\ln C_{2}$ and manipulate $(1)$ then I get
$$e^{-\frac{1}{x}}=y-1-y+C_{2}\quad\quad\quad\quad(2)$$ $$\iff e^{-\frac{1}{x}}+1=C_{2}$$ $$\Rightarrow\frac{1}{e}+1=C_{2}\quad (\textrm{for}\quad x=1)$$
which after plugging $C_{2}$ into $(2)$ gives
$$e^{-\frac{1}{x}}+1=e^{-1}+1$$
which is obviously nonsense. Any help pointing out where I am going wrong is greatly appreciated.
I think you probably put wrong values. Plugging in we have $-1=\ln|-1|+c_1$ hence $c_1=-1$ which makes sense.