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Higher Order Derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsHigher Order Derivatives3 Marks
Question
If $\text{y}=\text{e}^\text{x}(\sin\text{x}+\cos\text{x})$ prove that $\frac{\text{d}^2\text{y}}{\text{dx}^2}-2\frac{\text{dy}}{\text{dx}}+2\text{y}=0$
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Answer

$\text{y}=\text{e}^\text{x}(\sin\text{x}+\cos\text{x})$
Differentiating w.r.t.x, we get
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{e}^\text{x}(\cos\text{x}-\sin\text{x})+(\sin\text{x}+\text{cos}\text{x})\text{e}^\text{x}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{y}+\text{e}^\text{x}(\cos\text{x}-\sin\text{x})$
Differentiating w.r.t.x, we get
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{\text{dy}}{\text{dx}}+\text{e}^\text{x}(-\sin\text{x}-\cos\text{x})+(\cos\text{x}-\sin\text{x})\text{e}^\text{x}$
$=\frac{\text{dy}}{\text{dx}}-\text{y}+(\cos\text{x}-\sin\text{x})\text{e}^\text{x}$
Adding and substracting y on RHS
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{\text{dy}}{\text{dx}}+\text{e}^\text{x}(-\sin\text{x}-\cos\text{x})+(\cos\text{x}-\sin\text{x})\text{e}^\text{x}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}-2\frac{\text{dy}}{\text{dx}}+2\text{y}=0$

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