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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS5 Marks
Question
If $y = (\tan^{–1}x)^2,$ show that $\text{(x}^{2}+\text{1})^{2}\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}+\text{2x}(\text{x}^{2}+{1)}\frac{\text{dy}}{\text{dx}}=2.$
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Answer

$\text{y}=(\tan^{-1}\text{x})^{2}\Rightarrow\frac{\text{dy}}{\text{dx}}=2\tan^{-1}\text{x}\cdot\frac{\text{1}}{\text{1+x}^{2}}.$
$\Rightarrow{(1}+\text{x}^{2})\frac{\text{dy}}{\text{dx}}=\text{2 tan}^{-1}\text{x}$
$\therefore\text{(1+x}^{2})\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}+\text{2x}\cdot\frac{\text{dy}}{\text{dx}}=\frac{\text{2}}{\text{1+x}^{2}}$
$\Rightarrow\text{(1+x}^{2})\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}+\text{2x}\text{ (1+x}^{2})\frac{\text{dy}}{\text{dx}}=2.$

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