Using substitution to reduce a differential equation to a known form

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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:

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

Solve the following:

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

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