Solve the initial value problem: $\frac{dy}{dx} = e^{x+y}$, given $y(0)=0$.

You are correct, and their solution is wrong. Their solution indeed satisfies the differential equation - however, ${y(0)}$ for their solution ${\neq 0}$. In fact, ${y(x=0)}$ doesn't even exist for their solution, since the function blows up to infinity