Form the differential equation by eliminating $A, B$ from $y = e^{2 x} (A cos 3 x + B sin 3 x)$ is

\frac{d^{2} y}{d x^{2}} + 4 \frac{d y}{d x} - 13 y = 0

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\frac{d^{2} y}{d x^{2}} - 4 \frac{d y}{d x} - 13 y = 0

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\frac{d^{2} y}{d x^{2}} - 4 \frac{d y}{d x} + 13 y = 0

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\frac{d^{2} y}{d x^{2}} + 4 \frac{d y}{d x} + 13 y = 0

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Solution

The correct option is D $\frac{d^{2} y}{d x^{2}} - 4 \frac{d y}{d x} + 13 y = 0$
$y = e^{2 x} (A cos 3 x + B sin 3 x)$
$\frac{d y}{d x} = 2 e^{2 x} (A cos 3 x + B sin 3 x) + e^{2 x} (- 3 A sin 3 x + 3 B cos 3 x)$
$\frac{d y}{d x} = 2 y + e^{2 x} (- 3 A sin 3 x + 3 B cos 3 x)$
$\frac{d^{2} y}{d x^{2}} = 2 \frac{d y}{d x} + 2 e^{2 x} (- 3 A sin 3 x + 3 B cos 3 x) + e^{2 x} (- 9 A cos 3 x - 9 B sin 3 x)$
$\frac{d^{2} y}{d x^{2}} = 2 \frac{d y}{d x} + 2 (\frac{d y}{d x} - 2 y) - 9 y$
$\frac{d^{2} y}{d x^{2}} = 2 \frac{d y}{d x} + 2 \frac{d y}{d x} - 13 y$
$\frac{d^{2} y}{d x^{2}} = 4 \frac{d y}{d x} + 13 y = 0$

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