So, I'm trying to solve the following differential equation methodically:
$$ y(x)-2-\int_0^x e^{y(t)-t} \, dt = 0 $$
I rearranged the equation a bit and differentiated both sides and got:
$$ e^x \cdot y'(x)=e^{y(x)} $$
Now, I immediately see that a possible solution is $ y(x) = x $, but how do I solve this more methodically?
Separation of variables: $$ e^x \cdot y'=e^{y} $$ $$ e^x \cdot \frac{dy}{dx}=e^{y} $$ $$ e^{-y} \cdot \frac{dy}{dx}=e^{-x} $$ $$\int e^{-y} \, dy=\int e^{-x} \, dx$$