Differential Equations Question 291
Question: The solution of $ \frac{dy}{dx}=\sqrt{1-x^{2}-y^{2}+x^{2}y^{2}} $ is
Options:
A) $ si{n^{-1}}y=si{n^{-1}}x+c $
B) $ 2si{n^{-1}}y=\sqrt{1-x^{2}}+si{n^{-1}}x+c $
C) $ 2si{n^{-1}}y=x\sqrt{1-x^{2}}+si{n^{-1}}x+c $
D) $ 2si{n^{-1}}y=x\sqrt{1-x^{2}}+{{\cos }^{-1}}x+c $
Show Answer
Answer:
Correct Answer: D
Solution:
[c] $ \because \frac{dy}{dx}=\sqrt{1-x^{2}-y^{2}+x^{2}y^{2}} $
$ \frac{dy}{dx}=\sqrt{(1-x^{2})(1-y^{2});}\Rightarrow \frac{dy}{\sqrt{1-y^{2}}}=\sqrt{1-x^{2}}.dx $
$ \Rightarrow \int{\frac{dy}{\sqrt{1-y^{2}}}=\int{\sqrt{1-x^{2}}.dx}} $ [integrating b/s]
$ \Rightarrow {{\sin }^{-1}}( \frac{y}{1} )=\frac{x}{2}\sqrt{1-x^{2}}+\frac{1}{2}{{\sin }^{-1}}( \frac{x}{1} )+c $
$ \Rightarrow 2{{\sin }^{-1}}y=x\sqrt{1-x^{2}}+{{\sin }^{-1}}x+c $ where c is an arbitrary constant.
BETA
