The complementary function of $(D^{2} + 1) y = e^{2 x}$ is :

(A x + B) e^{x}

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A cos x + B sin x

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(A x + B) e^{2 x}

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(A x + B) e^{- x}

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Solution

The correct option is C $A cos x + B sin x$
Auxillary equation of above differential equation is $(D^{2} + 1) y = 0$ .
$\Rightarrow \frac{d^{2} y}{d x^{2}} + y = 0$
$\Rightarrow y = A cos x + B sin x$

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