Question

Solve the differential equation \[ \frac{dy}{dx} = e^{x + y}. \]

Hint:

Separate variables when the differential equation is separable and integrate both sides.

Answer 1

Rewrite the equation: \[ \frac{dy}{dx} = e^x \cdot e^y. \] Separate variables: \[ \frac{dy}{e^y} = e^x \, dx. \] Rewrite as: \[ e^{-y} dy = e^x dx. \] Integrate both sides: \[ \int e^{-y} dy = \int e^x dx. \] Calculate integrals: \[ - e^{-y} = e^x + C. \] Rearranged, \[ e^{-y} = - e^x + C', \] where \(C' = -C\). Or, \[ \boxed{ e^{-y} + e^x = C. } \]