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Question

(C) Given the family of curves: $y = c_1 e^{c_2 x} \dots (i)$Differentiating with respect to $x$,we get:$y' = c_1 c_2 e^{c_2 x} = c_2 y \dots (ii)$Aga

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$y y'' = (y')^2$

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(C) Given the family of curves: $y = c_1 e^{c_2 x} \dots (i)$Differentiating with respect to $x$,we get:$y' = c_1 c_2 e^{c_2 x} = c_2 y \dots (ii)$Again,differentiating with respect to $x$,we get:$y'' = c_2 y' \dots (iii)$From equation $(ii)$,we have $c_2 = \frac{y'}{y}$.Substituting this value of $c_2$ into equation $(iii)$:$y'' = \left( \frac{y'}{y} \right) y'$Multiplying both sides by $y$:$y y'' = (y')^2$This is the required differential equation.

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:

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