Question

Solve the differential equation. dz/dt + e^(t + z) = 0

Answer 1

$\frac{dz}{dt} + e^{t + z} = 0 \ \Rightarrow\ \frac{dz}{dt} = - e^{t + z} \ \Rightarrow\ \frac{dz}{dt} = - e^{t}e^z \ \Rightarrow\ \\ \\ \frac{dz}{e^z} = -e^{t} dt \ \Rightarrow\ \int e^{-z} dz=\int -e^{t} dt \ \Rightarrow\ \\ \\ -e^{-z} = -e^{t} + C \ \Rightarrow\ e^{-z} = e^{t} - C \ \Rightarrow\ -z = \ln(e^{t} - C)\ \Rightarrow\ \\ \\ z = -\ln(e^{t} - C)$