Solve the nonlinear DE $y'' + yy' = 0$ with $y(0) = 1, y'(0) = -1$ and find an interval of definition for this solution.
My progress: After substitution $u = y' = \frac{dy}{dx}$, I found $\frac{-y^2}{2} + C = u$ where $C$ is constant. However, the sign of $C$ makes the indefinite integral unclear, unsure what to consider. Moreover, I did not understand what does "interval of definition" mean. Any help on how to continue in such circumstances and elaboration on "interval of definition" would be welcomed.
Plugging in your initial conditions, you get that
$$-\frac{1}{2} + C = -1 \implies C = -\frac{1}{2}$$
From here just use separation of variables again
$$\int -dx = \int \frac{2\:dy}{1+y^2} \implies y = -\tan\left(\frac{x}{2}+C\right)$$
with another $+C$. Again plugging in initial conditions gives us $C= -\frac{\pi}{4}$. $y$ is only defined when $x-\frac{\pi}{2}$ is between odd multiples of $\pi$ (why?). Can you figure out which interval that would have to be for this problem?