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(C) Given the family of curves $y = C_1 e^{C_2 x}$.First,differentiate with respect to $x$:$y^{\prime} = C_1 C_2 e^{C_2 x}$Since $y = C_1 e^{C_2 x}$,w

Vedclass · Accepted Answer

$y y^{\prime \prime} = (y^{\prime})^2$

Vedclass · Answer

(C) Given the family of curves $y = C_1 e^{C_2 x}$.First,differentiate with respect to $x$:$y^{\prime} = C_1 C_2 e^{C_2 x}$Since $y = C_1 e^{C_2 x}$,we can substitute this into the derivative:$y^{\prime} = y C_2$$C_2 = \frac{y^{\prime}}{y}$Now,differentiate $y^{\prime} = C_1 C_2 e^{C_2 x}$ again with respect to $x$:$y^{\prime \prime} = C_1 C_2^2 e^{C_2 x}$Substitute $y = C_1 e^{C_2 x}$ into the second derivative:$y^{\prime \prime} = (C_1 e^{C_2 x}) C_2^2 = y C_2^2$Substitute $C_2 = \frac{y^{\prime}}{y}$ into the equation:$y^{\prime \prime} = y \left( \frac{y^{\prime}}{y} \right)^2$$y^{\prime \prime} = y \frac{(y^{\prime})^2}{y^2}$$y^{\prime \prime} = \frac{(y^{\prime})^2}{y}$$y y^{\prime \prime} = (y^{\prime})^2$Thus,the correct option is $C$.

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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