For the equation $x^{''} (t) + 3 x^{'} (t) + 2 x (t) = 5$ , the solution $x (t)$ approaches to the following values as $t \to \infty$

0

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5 / 2

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5

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10

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Solution

The correct option is B $5 / 2$
$x^{''} (t) + 3 x^{'} (t) + 2 x (t) = 5$
A.E. is $D^{2} + 3 D + 2 = 0$
$D = - 1, - 2$
C.F. = $C_{1} e^{- t} + C_{2} e^{- 2 t}$
P.I = $\frac{1}{D^{2} + 3 D + 2} 5 = \frac{5}{2}$
$x (t) = C_{1} e^{- t} + C_{2} e^{- 2 t} + \frac{5}{2}$
$\Rightarrow$ $x (t \to \infty) = \frac{5}{2}$

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