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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations3 Marks
Question
Form the differential equation having $\text{y}=(\sin^{-1}\text{x})+\text{A}\cos^{-1}\text{x}+\text{B}$ where A and B are aribitrary constants, as its general solution.
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Answer

$\text{y}=(\sin^{-1}\text{x})+\text{A}\cos^{-1}\text{x}+\text{B}$

$\frac{\text{dy}}{\text{dx}}=2\sin^{-1}\text{x}\times\Big(\frac{1}{\sqrt{1-\text{x}^2}}\Big)+\text{Ax}\Big(\frac{-1}{\sqrt{1-\text{x}^2}}\Big)=0$

$\sqrt{1-\text{x}^2}\frac{\text{dy}}{\text{dx}}=2\sin^{-1}\text{x}-\text{A}$

$\sqrt{1-\text{x}^2}\frac{\text{d}^2\text{y}}{\text{dx}^2}+\frac{\text{dy}}{\text{dx}}\Big(\frac{1}{\sqrt{1-\text{x}^2}}\Big)(-2\text{x})\Big(\frac{1}{\sqrt{1-\text{x}^2}}\Big)-0$

$(1-\text{x}^2)\frac{\text{d}^2\text{y}}{\text{dx}^2}-2\text{x}\frac{\text{dy}}{\text{dx}}-2=0$

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