Introduction to Differential equations

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Question

The degree of the differential equation which has a solution $y^{2} = 4 a (x + a^{2})$ where a is arbitrary constant

Moderate

Solution

Differentiating $y^{2} = 4 a (x + a^{2})$
We have, $2 y \frac{d y}{d x} = 4 a \Rightarrow a = \frac{y y^{'}}{2}$
Putting in $(1)$
$\begin{array}{l} y^{2} = 4 \frac{y y^{'}}{2} (x + \frac{1}{4} y^{2} y^{2}) \\ = 2 x y y^{'} + y^{3} y^{' 3} \end{array}$
So the degree is 3

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