Not to be confused with homogeneous equations, an equation homogeneous in x and y of degree n is an equation of the form
Such that
.
Then the equation can take the form
Which is essentially another in the form
.
If we can solve this equation for
, then we can easily use the substitution method mentioned earlier to solve this equation. Suppose, however, that it is more easily solved for
,
So that
.
We can differentiate this to get
Then re-arranging things,
So that upon integrating,
We get
Thus, if we can eliminate y' between two simultaneous equations
and
,
then we can obtain the general solution..
A function P is homogeneous of order
if
. A homogeneous ordinary differential equation is an equation of the form P(x,y)dx+Q(x,y)dy=0 where P and Q are homogeneous of the same order.
The first usage of the following method for solving homogeneous ordinary differential equations was by Leibniz in 1691. Using the substitution y=vx or x=vy, we can make turn the equation into a separable equation.



Now we need to find v':

Plug back into the original equation


- Solve for v(x), then plug into the equation of v to get y

Again, don't memorize the equation. Remember the general method, and apply it.

Let's use
. Solve for


Now plug into the original equation



Solve for v







Plug into the definition of v to get y.



We leave it in
form, since solving for y would lose information.
Note that there should be a constant of integration in the general solution. Adding it is left as an exercise.

Lets use
again. Solve for


Now plug into the original equation



Solve for v:



Use the definition of v to solve for y.


An equation that is a function of a quotient of linear expressions
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Given the equation
,
We can make the substitution x=x'+h and y=y'+k where h and k satisfy the system of linear equations:

Which turns it into a homogeneous equation of degree 0: