The differential equation of the family y=a e | vedclass

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(D) Given the family of curves: $y = (a + bx + cx^2)e^x$  This can be written as $y e^{-x} = a + bx + cx^2$.  Differentiating with respect to $x$ thre

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$y^{\prime \prime \prime}-3 y^{\prime \prime}+3 y^{\prime}-y=0$

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(D) Given the family of curves: $y = (a + bx + cx^2)e^x$  This can be written as $y e^{-x} = a + bx + cx^2$.  Differentiating with respect to $x$ three times:  First derivative: $\frac{d}{dx}(y e^{-x}) = b + 2cx$  $\Rightarrow y' e^{-x} - y e^{-x} = b + 2cx$  Second derivative: $\frac{d}{dx}(y' e^{-x} - y e^{-x}) = 2c$  $\Rightarrow y'' e^{-x} - y' e^{-x} - (y' e^{-x} - y e^{-x}) = 2c$  $\Rightarrow y'' e^{-x} - 2y' e^{-x} + y e^{-x} = 2c$  Third derivative: $\frac{d}{dx}(y'' e^{-x} - 2y' e^{-x} + y e^{-x}) = 0$  $\Rightarrow y''' e^{-x} - y'' e^{-x} - 2(y'' e^{-x} - y' e^{-x}) + (y' e^{-x} - y e^{-x}) = 0$  $\Rightarrow y''' - y'' - 2y'' + 2y' + y' - y = 0$  $\Rightarrow y''' - 3y'' + 3y' - y = 0$  Thus, the correct option is $D$.

The differential equation of the family $y=a e^x+b x e^x+c x^2 e^x$ of curves, where $a, b, c$ are arbitrary constants, is

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