Consider the following equation $ $ y”(x) = -2e^{-y}. $ $

The following code

`DSolve[y''[x] == -2 Exp[-y[x]], y[x], x] //FullSimplify `

returns

`{{y[x] -> Log[(2 (-1 + Cosh[Sqrt[C[1]] (x + C[2])]))/C[1]]}, {y[x] -> Log[(2 (-1 + Cosh[Sqrt[C[1]] (x + C[2])]))/C[1]]}} `

(They the same solution. Let’s ignore that first.) If I impose an initial condition $ y(0) = 0$ then Mathematica fails to return a solution

`DSolve[{y''[x] == -2 Exp[-y[x]], y[0] == 0}, y[x], x] `

with the error message

DSolve::bvfail: For some branches of the general solution, unable to solve the conditions.

But a solution does exist. One can choose $ C[1] = 1$ and $ C[2] = \cosh^{-1}(3/2)$ in the solution and verify that $ y(0) = 0$ .

Any idea why is this the case?