Supposedly, it is possible to determine information about the constants of this IVP solution, without computing the solution of the differential equation. Here's how I solve this.
Let $z=y'$. In brief, we then have the separable equation $$z\frac{dz}{dy}=f(y,z)=e^{2y}$$
and solving for $z$ $$z=\sqrt{e^{2y}+c}$$
which can be made into a separable equation and integrated. After playing around with the initial conditions, I did find that $c=0$. But this is a contradiction with the results of the second separable equation. Can we really solve $c$ before finding the explicit form of $y$?
Best Answer
From my answer to this, we see that we need to take the $+$ of the square root and the answer will be case $3$ where $C_1=0$. Thus the answer is
$$y=-\log(1-x)$$