Final answer:
The given differential equation, (x - y) dx + x dy = 0, is a homogeneous DE that can be solved through substitution and separation of variables, resulting in an equation that can be easily integrated to find the solution.
Step-by-step explanation:
We are going to solve the given homogeneous differential equation (DE): (x - y) dx + x dy = 0. A common technique is to use a substitution that will reduce this equation to a separable form. We'll take the approach of substituting v = y/x, which implies that y = vx. Taking the derivative, we get dy/dx = v + x dv/dx. We can then substitute these back into the original DE.
Substituting the values into the DE, we get:
- (x - vx) dx + x (v + x dv/dx) dy = 0
Now, this simplifies to:
- (1 - v) dx + x( dv + v dx ) = 0
Separating the variables and integrating both sides gives us the solution.
The substitution and separation of variables are key steps in solving this differential equation effectively.