Homogeneous differential equation
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A homogeneous differential equation has several distinct meanings.
One meaning is that a first-order ordinary differential equation is homogeneous if it has the form
To solve such equations, one makes the change of variables u = y/x, which will transform such an equation into separable one.
A similarly called object, the linear homogeneous differential equation, is a differential equation which is linear and equal to 0.
[edit] Example of deriving a homogenous equation
A well known homogenous equation in x and y of degree m, subsequently showing one of Euler's identities is as follows.
Deriving We obtain the following,
.
Where denotes the first derivative of F with respect to the homogenous argument.
Also,
Now taking each derivative and multiplying by its corresponding variable we arrive at the following equation.
Which in turn is one of Euler's identities,
This identity is generalized by Euler's theorem on homogeneous functions.