Let x = x(y) be the solution of the differential equation y = (x-y dy/dx) sin (x/y), y > 0 and x(1) = π/2.

Question

Let x = x(y) be the solution of the differential equation \(y=\left(x-y \frac{d x}{d y}\right) \sin \left(\frac{x}{y}\right), y>0\) and \(x(1)=\frac{\pi}{2}.\)
Then \(\cos (x(2))\) is equal to :

(1) \(1-2\left(\log _{\mathrm{e}} 2\right)^{2}\)

(2) \(2\left(\log _{\mathrm{e}} 2\right)^{2}-1\)

(3) \(2\left(\log _{e} 2\right)-1\)

(4) \(1-2\left(\log _{e} 2\right)\)

Answer 1

Correct option is (2) \(2\left(\log _{\mathrm{e}} 2\right)^{2}-1\)   

\(y d y=(x d y-y d x) \sin \left(\frac{x}{y}\right)\)

\(\frac{d y}{y}=\left(\frac{x d y-y d x}{y^{2}}\right) \sin \left(\frac{x}{y}\right)\)

\(\frac{d y}{y}=\sin \left(\frac{x}{y}\right) d\left(-\frac{x}{y}\right)\)

\(\Rightarrow \quad \ell n y=\cos \frac{x}{y}+C\)

\( \mathrm{x}(1)=\frac{\pi}{2} \Rightarrow 0=\cos \frac{\pi}{2}+\mathrm{C} \Rightarrow \mathrm{C}=0\)

\(\ell \text { ny }=\cos \frac{x}{y} \)

\( \text { but } y=2 \Rightarrow \cos \frac{x}{2}=\ell n 2 \)

\(\cos \mathrm{x} =2 \cos ^2 \frac{x}{2}-1 \)

\(=2(\operatorname{\ell n} 2)^2-1\)