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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS5 Marks
Question
$\text{If}\ \text{y}=\sin(\sin\text{x}),\ \text{prove that}\frac{\text{d}^2\text{y}}{\text{dx}^2}+\tan\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}\cos^2\text{x}=0.$
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Answer

$\text{y}=\sin(\sin\text{x})$
$\frac{\text{dy}}{\text{dx}}=\cos(\sin\text{x})\cos\text{x}$
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\cos(\sin\text{x})(-\sin\text{x})+\cos^2\text{x}[-\sin(\sin\text{x})]$
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=-\sin\text{x}\cos(\sin\text{x})-\cos^2\text{x}\sin(\sin\text{x})$
$\text{L.H.S}=\frac{\text{d}^2\text{y}}{\text{dx}^2}+\tan\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}\cos^2\text{x}$
$=-\sin\text{x}\cos(\sin\text{x})-\cos^2\text{x}\sin(\sin\text{x})\\+\tan\text{x}\cos\text{x}\cos(\sin\text{x})+\cos^2\text{x}\sin\text{x}(\sin\text{x})$
$=-\sin\text{x}\cos(\sin\text{x})+\frac{\sin\text{x}}{\cos\text{x}}\cos\text{x}\cos(\sin\text{x})$
$=-\sin\text{x}\cos(\sin\text{x})+\sin\text{x}\cos(\sin\text{x})$
$=0=\text{R.H.S.}$

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