JEE Main · 2009 · 12 Apr 2009 · Offline · Differential Equations

The differential equation which represents the family of curves

y = c_{1} e^{c_{2} x}

, where

c_{1}

and

c_{2}

are arbitrary constants is

Answer: (D)

Solution

Differentiating

y = c_{1} e^{c_{2} x}

gives

y^{'} = c_{2} c_{1} e^{c_{2} x} = c_{2} y

, and differentiating once more gives

y^{''} = c_{2} y^{'}

. Eliminating the constant through

c_{2} = \frac{y ^{'}}{y}

turns the second relation into

y^{''} = \frac{y ^{'}}{y} y^{'} = \frac{( y ^{'} ) ^{2}}{y}

, so the family satisfies

y y^{''} = (y^{'})^{2}

JEE Main · 2009 · 12 Apr 2009 · Offline · Single correct option

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