1st Order Homogeneous Differential Equations of the Form (y/x)


Given a homogeneous 1st order differential equation of the form:


...where F is a function of y/x

...we can make a substitution via the following:


$$y=vx $$

...and via the product rule:

$$\frac{dy}{dx}=v'x+x'v\;,\;x'=1 $$


$$\frac{dy}{dx}=v'x+v $$

(Ex 1) Find the general solution of the following:

$$\frac{dy}{dx}=\frac{x^2+3y^2}{2xy} $$

First, attempt to make the right side of the equation a function solely of y/x. Upon inspection you may realize that it is possible to do this by dividing both the numerator and denominator by x-squared.

$$\frac{dy}{dx}=\frac{\frac{x^2}{x^2}+\frac{3y^2}{x^2}}{\frac{2xy}{x^2}} $$

...which becomes:

$$\frac{1+3(\frac{y}{x})^2}{2(\frac{y}{x})} $$

Now, let:



...and make the appropriate substitutions:


The above equation is now separable. Go ahead and solve:





Since C is an arbitrary constant at this point, we can write:


...and recalling one of our log rules, the above can be written as:


...which becomes:


Remember that we are trying to find a general solution for y. So at this point we can go ahead and back-substitute for v in the above equation:





...and finally, the general solution to our original differential equation is represented by:

$$y=\pm \sqrt{Cx^3-x^2}$$