Often the use of a substitution can alter the format of a first order differential equation and convert it into a form that is familiar.

Solve the following:

  1. \left( {x^2} - {y^2} \right) \dfrac{dy}{dx} - xy = 0 \hphantom{aa}\text{ by using the substitution } \hphantom{a} z = \dfrac{y}{x}

Solve the following:

  1. 2(1+x^2) \dfrac{dy}{dx} + 2xy = \dfrac{1}{y} \hphantom{aa}\text{ by using the substitution } \hphantom{a} z = y^2