Differential equation representing the family of curves $y = e^{x} (A c o s x + B s i n x)$ is $\frac{d^{2} y}{d x^{2}} - 2 \frac{d y}{d x} + 2 y = 0$

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Solution

True
Given that, $y = e^{x} (A c o s x + B s i n x)$
On differentiating w.r.t. x, we get
$\frac{d y}{d x} = e^{x} (- A s i n x + B c o s x) + e^{x} (A c o s x + B s i n x) \Rightarrow \frac{d y}{d x} - y = e^{x} (- A s i n x + B c o s x)$
Again differentiating w.r.t. x, we get
$\frac{d^{2} y}{d x^{2}} - \frac{d y}{d x} = e^{x} (- A c o s x - B s i n x) + e^{x} (- A s i n x + B c o s x) \Rightarrow \frac{d^{2} y}{d x^{2}} - \frac{d y}{d x} + y = \frac{d y}{d x} - y \Rightarrow \frac{d^{2} y}{d x^{2}} - 2 \frac{d y}{d x} + 2 y = 0$

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