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प्रश्न
If y = ex (sin x + cos x) prove that \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\] ?
उत्तर
Here,
\[y = e^x \left( \sin x + \cos x \right)\]
\[\text { Differentiating w . r . t . x, we get }\]
\[\frac{d y}{d x} = e^x \left( \sin x + \cos x \right) + e^x \left( \cos x - \sin x \right) = 2 e^x \cos x\]
\[\text { Differentiating w . r . t . x, we get }\]
\[\frac{d^2 y}{d x^2} = 2 e^x \cos x - 2 e^x \sin x\]
\[\text { Now, }\]
\[\text { LHS }= \frac{d^2 y}{d x^2} - 2\frac{d y}{d x} + 2y\]
\[ = 2 e^x \cos x - 2 e^x \sin x - 4 e^x \cos x + 2 e^x \left( \sin x + \cos x \right)\]
\[ = 0 = \text { RHS }\]
Hence proved.
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