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Roughly speaking, '''Euler's homogeneous function theorem''' asserts that the positively homogeneous functions of a given degree are exactly the solution of a specific [[partial differential equation]]. More precisely:
{{Math theorem
{{Math theorem|name=Euler's homogeneous function theorem |math_statement=If {{mvar|f}} is a [[partial function|(partial) function]] of {{mvar|n}} real variables that is positively homogeneous of degree {{mvar|k}}, and [[continuously differentiable]] in some open subset of <math>\R^n,</math> then it satisfies in this open set the [[partial differential equation]]▼
| name = Euler's homogeneous function theorem
▲
<math display="block">k\,f(x_1, \ldots,x_n)=\sum_{i=1}^n x_i\frac{\partial f}{\partial x_i}(x_1, \ldots,x_n).</math>
Conversely, every maximal continuously differentiable solution of this partial differentiable equation is a positively homogeneous function of degree {{mvar|k}}, defined on a positive cone (here, ''maximal'' means that the solution cannot be prolongated to a function with a larger ___domain).
}}
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