If a and b are constants, the most general solution of the differential equation $\frac{d^{2} x}{d t^{2}} + 2 \frac{d x}{d t} + x = 0$ is

a e^{- t}

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a e^{- t} + b t e^{- t}

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a e^{t} + b t e^{- t}

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a e^{- 2 t}

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Solution

The correct option is B $a e^{- t} + b t e^{- t}$
$\frac{d^{2} x}{d t^{2}} + 2 \frac{d x}{d t} + x = 0$
Auxiliary equation is $D^{2} + 2 D + 1 = 0$
$D = - 1, - 1$
Sol. is $x = a e^{- t} + b t e^{- t}$

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