Question

Solve the differential equation \[ (x + y) \, dy + (x - y) \, dx = 0, \quad \text{if} \quad y = 1 \text{ when } x = 1. \] 

Hint:

When solving a first-order differential equation with an initial condition, first solve for the general solution and then substitute the initial condition to find the particular solution.

Answer 1

Step 1: Rearrange the terms of the equation: \[ \frac{dy}{dx} = \frac{-(x - y)}{x + y}. \] 

Step 2: Use substitution, let \( v = x + y \), so that \( dv = dx + dy \). 

Step 3: Substitute into the equation and solve for \( y \). 

Step 4: Apply the initial condition to find the particular solution.