Form the differential equation corresponding to the family of curves $y = c (x - c)^{2}$ , where c is a parameter.

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Solution

We have $y = c (x - c)^{2}$ .........(1)
Differentiate both sides, $y^{'} = 2 c (x - c)$ .......(2)
Divide (2) by (1)
$\Rightarrow \frac{y}{y^{'}} = \frac{x - c}{2}$
$\Rightarrow x - c = \frac{2 y}{y^{'}}$
$\Rightarrow c = x - \frac{2 y}{y^{'}}$
Now substitute the value of $c$ (2)
$\Rightarrow y^{'} = 2 (x - \frac{2 y}{y^{'}}) \cdot \frac{2 y}{y^{'}}$
$\Rightarrow (y^{'})^{3} = 4 y (x y^{'} - 2 y)$

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