Family of curves $y = e^{x} (A cos x + B sin x)$ , represents the differential equation?

\frac{d^{2} y}{d x^{2}} = 2 \frac{d y}{d x} - y

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\frac{d^{2} y}{d x^{2}} = 2 \frac{d y}{d x} - 2 y

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\frac{d^{2} y}{d x^{2}} = \frac{d y}{d x} - 2 y

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None of the above

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Solution

The correct option is B $\frac{d^{2} y}{d x^{2}} = 2 \frac{d y}{d x} - 2 y$
$y = e^{x} (A c o s x + B s i n x)$
$y^{1} = e^{x} (- A s i n x + B c o s x)$ $+ (A c o s x + B s i n x) e^{x}$
$\Rightarrow y^{1} = A (c o s x - s i n x) e^{x} + B (s i n x + c o s x) e^{x}$
& $y^{11} = A [(- s i n x - c o s x) e^{x} + (c o s x - s i n x) e^{x}] + B [(c o s x - s i n x) e^{x} + (s i n x + c o s x) e^{x}]$
$\Rightarrow y^{11} = A (- 2 s i n x) e^{x} + B (2 c o s x) e^{x}$
$y^{11} = + 2 y^{1} - 2 y$
$\frac{d^{2} y}{{d x}^{2}} = \frac{2 d y}{d x} - 2 y$

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