Solving Differential Equation (dz/dt) + e^(t+z) = 0 with Arbitrary Constant C

[cgward]

I have to solve the differential equation and let C represent an arbitrary constant.

(dz/dt)+e^(t+z)=0

I can't seem to figure it out i wind up getting z=-2e^(t^2)+e^(C)

 

[Matthew Rodman]

Ok, divide by e^z on both sides to get

e^{-z} \frac{dz}{dt} = -e^t

then integrate with respect to t,

-e^{-z} = k - e^t

re-arrange to get

z(t) = - \ln{(k + e^t)}.

(k is a constant).