If the right side of the equation dy dx = f(x, y) can be expressed as a function of the ratio y/x only, then the equation is said to be homogeneous. such equations can always be transformed into separable equations by a change of the dependent variable. the following method outline can be used for any homogeneous equation. that is, the substitution y = xv(x) transforms a homogeneous equation into a separable equation. the latter equation can be solved by direct integration, and then replacing v by y x gives the solution to the original equation. dy/dx = (x^2 + 5y^2)/ 2xy

 y = x*sqrt(Cx - 1)

Step-by-step explanation:

Given:

                                  dy / dx = (x^2 + 5y^2) / 2xy

Find:

Solve the given ODE by using appropriate substitution.

Solution:

- Rewrite the given ODE:

                               dy/dx = 0.5(x/y) + 2.5(y/x)

- use substitution y = x*v(x)

                               dy/dx = v + x*dv/dx

- Combine the two equations:

                                v + x*dv/dx = 0.5*(1/v) + 2.5*v

                                x*dv/dx = 0.5*(1/v) + 1.5*v

                                x*dv/dx = (v^2 + 1) / 2v

-Separate variables:

                                 (2v.dv / (v^2 + 1) = dx / x  

- Integrate both sides:

                                 Ln (v^2 + 1) = Ln(x) + C

                                 v^2 + 1 = Cx

                                 v = sqrt(Cx - 1)

- Back substitution:

                                (y/x) = sqrt(Cx - 1)

                               y = x*sqrt(Cx - 1)