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Q.

Form the differential equation by eliminating a, b from $y = {ae}^{3 x} + {be}^{4 x}$

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a

$\frac{d^{2} y}{d x^{2}} + 7 \frac{d y}{d x} + 12 y = 0$

b

$\frac{d^{2} y}{d x^{2}} - 7 \frac{d y}{d x} - 12 y = 0$

c

$\frac{d^{2} y}{d x^{2}} - 7 \frac{d y}{d x} + 12 y = 0$

d

$\frac{d^{2} y}{d x^{2}} + 7 \frac{d y}{d x} - 12 y = 0$

answer is C.

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Detailed Solution

$y = a e^{3 x} + b e^{4 x} \Rightarrow \frac{d y}{d x} = 3 a e^{3 x} + 4 b e^{4 x} \Rightarrow \frac{d^{2} y}{d x^{2}} = 9 a e^{3 x} + 16 b e^{4 x} \Rightarrow \frac{d^{2} y}{d x^{2}} - (3 + 4) \frac{d y}{d x} + (3 \cdot 4) y = 0 \Rightarrow \frac{d^{2} y}{d x^{2}} - 7 \frac{d y}{d x} + 12 y = 0$

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