Ordinary Differential Equations/Homogeneous x and y

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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 y', 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



then we can obtain the general solution..

Homogeneous Ordinary Differential Equations[edit | edit source]

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.

Example 2[edit | edit source]

Let's use . Solve for y'(x,v,v')

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.

Example 3[edit | edit source]

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[edit | edit source]

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: