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Chain Rule of Differentiation

Form the diff...

Question

Form the differential equation
$y = a (x - a)^{2}$

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Solution

$y = a (x - a)^{2}$ _(1)

$y = 2 a (x - a)$ _(2)

Dividing both the equations, we get

$\Rightarrow \frac{y}{y^{'}} = \frac{x - a}{2}$

$\Rightarrow \frac{2 y}{y^{'}} = (x - a)$

$\Rightarrow a = x - \frac{2 y}{y^{'}}$

Substituting this in equation (1) and simplifying, we get

$y (y^{'})^{3} = x^{2} y^{'} (y^{'} - 1) + 2 y (x - 2 y)$

is the required equation.

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