Differential equations/Homogeneous differential equations

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HomogeneousEdit

DefinitionEdit

The word “homogeneous” can mean different things depending on what kind of differential equation you’re working with. A homogeneous equation in this sense is defined as one where the following relationship is true:

 

SolutionEdit

The solution to a homogeneous equation is to:

  1. Use the substitution   where   is a substitution variable.
  2. Implicitly differentiate the above equation to get  .
  3. Replace   and   with these expressions.
  4. Solve for  .
  5. Substitute with the expression   Then solve for  .

The advantage of this method is that the function is in terms of 2 variables, but we simplify the equation by relating   and   to each other.