Differential equations/Change of variables

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In a differential equation, if a certain term appears many different times, a substitution can be made similar to a  -substitution.


  1. Substitute a term for a variable (e.g.  ).
  2. Implicitly differentiate the variable (e.g.  ).
  3. Solve for the derivative that needs to be solved (e.g.  ).
  4. Solve the original equation in terms of   and then use the substitution for   to get the original equation back in terms of   and  .