Final answer:
To solve the given homogeneous differential equation (x-y)dx + xdy = 0, we can use the substitution u = x-y. The solution to the differential equation is y = -ln|x| + C.
Step-by-step explanation:
To solve the given differential equation (x-y)dx + xdy = 0, we can use the substitution u = x-y. Let's differentiate u with respect to x, du/dx = 1 - dy/dx.
By substituting u and du/dx into the original equation, we get (u+dy/dx)dx + x dy = 0. Simplifying this, we have u dx + x dy = 0.
Now we have a separate variable equation. Dividing both sides by x, we get (u/x) dx + dy = 0. Integrating both sides, ln|x| + y = C, where C is the constant of integration.
Solving for y, we have y = -ln|x| + C. This is the solution to the given homogeneous differential equation.