The differential equation satisfied by the system of parabolas y2 = 4a(x + a) is :

Question

Accepted Answer

${y^2} = 4ax + 4{a^2}$

differentiate with respect to x

$$ \Rightarrow 2y{{dy} \over {dx}} = 4a$$

$$ \Rightarrow a = \left( {{y \over 2}{{dy} \over {dx}}} \right)$$

So, required differential equation is

$${y^2} = \left( {4 \times {y \over 2}{{dy} \over {dx}}} \right)x + 4{\left( {{y \over 2}{{dy} \over {dx}}} \right)^2}$$

$$ \Rightarrow {y^2}{\left( {{{dy} \over {dx}}} \right)^2} + 2xy\left( {{{dy} \over {dx}}} \right) - {y^2} = 0$$

$$ \Rightarrow y{\left( {{{dy} \over {dx}}} \right)^2} + 2x\left( {{{dy} \over {dx}}} \right) - y = 0$$

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