a homogeneous differential equation dy g , consider the substitution v . Then, y xv and dy v x dv .Thus, the given differential equation becomes x dv f v v which is solved using variables separable method. This leads to the following result. Theorem .
If M x y dx N x y dy = is a homogeneous differential equation, then the change of variable vx , transforms into a separable equation in the variables v and x . Example . Solve x xydy We know that the given equation is homogeneous. Now, we rewrite the given equation as dy dx = Taking y vx , we have v x dv = v v or x dv v v Separating the variables, we obtain vdv v − = dx x .
On integration, we get log v − = log Hence v − = Cx , where C is an arbitrary constant. Now, replace v by y x to get y = Cx . Thus, we have y = Cx . Hence, y = ± Cx (or) y kx gives the general solution.
Example . Solve y xdy The given differential equation is homogeneous (verify!). Now, we rewrite the given equation in differential form dy Since the initial value of x is , we consider x > and take x . We have dy + Let y vx .
Then, v x dv v v , which becomes x dv v . By separating variables, we have dv v Upon integration, we get log v v or v v xC + = Now, we replace v by y x , we get y Cx + = (or) y Cx gives the general solution of the given differential equation. To determine the value of C , we use the condition that y = when x = . So, we get C = .