The differential equation of the family y = aex + bx ex + cx2 ex of curves, where a, b, c are arbitrary constants, is y"’+ ay"+ by’+ cy = 0 then :

y = ae x + bxe x + cx 2 e x y = e αx ( a + bx + cx 2 e x ) ⇒ d 3 y dx 3 - 3 α d 2 y dx 2 + 3 α 2 dy dx - α 3 y = 0 y = e x ( a + bx + cx 2 e x ) ⇒ d 3 y dx 3 - 3 d 2 y dx 2 + 3 dy dx - y = 0

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The differential equation of the family y = ae^x + bx e^x + cx² e^x of curves, where a, b, c are arbitrary constants, is y"’+ ay"+ by’+ cy = 0 then :

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a = 3

a = –3

b = 3

c = 1

answer is B, C.

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Detailed Solution

$y = {ae}^{x} + {bxe}^{x} + {cx}^{2} e^{x} (y = e^{αx} (a + bx + {cx}^{2} e^{x}) \Rightarrow \frac{d^{3} y}{{dx}^{3}} - 3 α \frac{d^{2} y}{{dx}^{2}} + 3 α^{2} \frac{dy}{dx} - α^{3} y = 0) y = e^{x} (a + bx + {cx}^{2} e^{x}) \Rightarrow \frac{d^{3} y}{{dx}^{3}} - 3 \frac{d^{2} y}{{dx}^{2}} + 3 \frac{dy}{dx} - y = 0$

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