Homogeneous Equations

The differential equation

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is homogeneous if the function f(x,y) is homogeneous, that is-

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Check that the functions

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are homogeneous.

In order to solve this type of equation we make use of a substitution (as we did in case of Bernoulli equations). Indeed, consider the substitution tex2html_wrap_inline64 . If f(x,y) is homogeneous, then we have

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Since y' = xz' + z, the equation (H) becomes

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which is a separable equation. Once solved, go back to the old variable y via the equation y = x z.

Let us summarize the steps to follow:

(1)
Recognize that your equation is an homogeneous equation; that is, you need to check that f(tx,ty)= f(x,y), meaning that f(tx,ty) is independent of the variable t;
(2)
Write out the substitution z=y/x;
(3)
Through easy differentiation, find the new equation satisfied by the new function z.
You may want to remember the form of the new equation:

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(4)
Solve the new equation (which is always separable) to find z;
(5)
Go back to the old function y through the substitution y = x z;
(6)
If you have an IVP, use the initial condition to find the particular solution.

Since you have to solve a separable equation, you must be particularly careful about the constant solutions.

Example: Find all the solutions of

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Solution: Follow these steps:

(1)
It is easy to check that tex2html_wrap_inline96 is homogeneous;
(2)
Consider tex2html_wrap_inline64 ;
(3)
We have

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which can be rewritten as

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This is a separable equation. If you don't get a separable equation at this point, then your equation is not homogeneous, or something went wrong along the way.

(4)
All solutions are given implicitly by

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(5)
Back to the function y, we get

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Note that the implicit equation can be rewritten as

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[Differential Equations]
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Author: Mohamed Amine Khamsi
Last Update 6-23-98

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