Question

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

• A. \( e^x + e^{-y} = c \)
• B. \( e^x + e^y = c \)
• C. \( e^{-x} + e^y = c \)
• D. \( e^{-x} + e^{-y} = c \)

Hint:

For separable differential equations, rewrite the equation so that all terms involving \( x \) are on one side and all terms involving \( y \) are on the other side. Then integrate both sides.

Answer 1

The Correct Option is A

We are given the equation: \[ \frac{dx}{dy} = e^{x + y} \] This equation is separable. We can rewrite it as: \[ \frac{dx}{e^x} = e^y \, dy \] Now, integrate both sides: \[ \int \frac{1}{e^x} \, dx = \int e^y \, dy \] The left-hand side integrates to \( -e^{-x} \), and the right-hand side integrates to \( e^y \). So we have: \[ -e^{-x} = e^y + C \] or equivalently: \[ e^x + e^{-y} = c \] where \( c \) is a constant. Thus, the correct solution is: \[ \boxed{e^x + e^{-y} = c} \]