The differential equation of the family of cu | 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 $x$ thr

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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: $y' e^{-x} - y e^{-x} = b + 2cx \implies (y' - y) e^{-x} = b + 2cx$  Second derivative: $(y'' - y') e^{-x} - (y' - y) e^{-x} = 2c \implies (y'' - 2y' + y) e^{-x} = 2c$  Third derivative: $(y''' - 2y'' + y') e^{-x} - (y'' - 2y' + y) e^{-x} = 0$  Since $e^{-x} 
eq 0$,we have $y''' - 3y'' + 3y' - y = 0$.

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

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