First slide

Methods of solving first order first degree differential equations

Question

$Let y = y (x) be a solution of the differential equation, \sqrt{1 - x^{2}} \frac{dy}{dx} + \sqrt{1 - y^{2}} = 0, | x | < 1 . If$ $y (\frac{1}{2}) = \frac{\sqrt{3}}{2}, then y (\frac{- 1}{\sqrt{2}}) is equal to:$

Moderate

A

$- \frac{1}{\sqrt{2}}$
B

$\frac{\sqrt{3}}{2}$
C

$- \frac{\sqrt{3}}{2}$
D

$\frac{1}{\sqrt{2}}$

Solution

$\begin{array}{l} \frac{d y}{\sqrt{1 - y^{2}}} + \frac{d x}{\sqrt{1 - x^{2}}} = 0 \Rightarrow \sin^{- 1} y + \sin^{- 1} x = c \\ At x = \frac{1}{2}, y = \frac{\sqrt{3}}{2} \Rightarrow c = \frac{π}{2} \Rightarrow \sin^{- 1} y = \cos^{- 1} x \\ Hence y (\frac{1}{\sqrt{2}}) = \sin (\cos^{- 1} \frac{1}{\sqrt{2}}) = \frac{1}{\sqrt{2}} \end{array}$

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