If sin ( y x ) = log e | x | + 2 is the solution of the differential equation x cos ( y x ) d y d x = y cos ( y x ) + x and y ( 1 ) = 3 , then 2 is equal to [2024]

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Solution

Q.
If $\sin (\frac{y}{x}) = \log_{e} | x | + \frac{α}{2}$ is the solution of the differential equation $x \cos (\frac{y}{x}) \frac{d y}{d x} = y \cos (\frac{y}{x}) + x$ and $y (1) = \frac{π}{3}$ , then $α^{2}$ is equal to [2024]

1 4

2 3

3 9

4 12

Ans.
(2)

The given differential equation is,

$x \cos (\frac{y}{x}) \frac{d y}{d x} = y \cos (\frac{y}{x}) + x$ ... (i)

Put $y = v x \Rightarrow \frac{d y}{d x} = v + x \frac{d v}{d x}$

$∴$ (i) becomes,

$x \cos v (v + x \frac{d v}{d x}) = v x \cdot \cos v + x \Rightarrow x \frac{d v}{d x} = \sec v$

$\Rightarrow \int \cos v d v = \int \frac{d x}{x} \Rightarrow \sin v = \log | x | + c$

$\Rightarrow \sin \frac{y}{x} = \log | x | + c$

Now, $y (1) = \frac{π}{3} \Rightarrow \sin \frac{π}{3} = c \Rightarrow c = \frac{\sqrt{3}}{2}$

So, the solution of given differential equation is,

$\sin (\frac{y}{x}) = \log | x | + \frac{\sqrt{3}}{2}$

On comparing with given solution, we get

$α = \sqrt{3} \Rightarrow α^{2} = 3$ .

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