Solving First-Order Linear Differential Equation: dx/dy + x = e^(-y)

Published on February 24, 2026

Calculus

Solve the first-order linear differential equation dx/dy + x = e^(-y) using an integrating factor.

Question

Okay, let’s solve the given differential equation step by step.

The given differential equation is:

This is a first-order linear differential equation in the form:

where

and

To solve this, we first find the integrating factor, which is given by:

Now, we multiply the entire equation by the integrating factor

The left side of the equation is the derivative of the product

with respect to

Now, we integrate both sides with respect to

Finally, we solve for

Thus, the solution to the differential equation is:

Final Answer: The final answer is

• First-Order Linear Differential Equation

A first-order linear differential equation can be written in the form

, where

and

are functions of

. The solution is found by first calculating the integrating factor

, and then multiplying the equation by

to get

. Integrating both sides with respect to

gives

, and finally,