Question

Assertion: $x \sin x \frac{d y}{d x}+(x+x \cos x+\sin x) y$
$=\sin x , y \left(\frac{\pi}{2}\right)=1-\frac{2}{\pi} \Rightarrow \displaystyle\lim _{ x \rightarrow 0} y ( x )=\frac{1}{3}$
Reason: The differential equation is linear with integrating factor $x(1-\cos x)$

• A. Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
• B. Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
• C. Assertion is correct, Reason is incorrect
• D. Assertion is incorrect, Reason is correct.

Answer 1

Answer: (A)

$\frac{ dy }{ dx }+\left(\frac{1}{\sin x }+\cot x +\frac{1}{ x }\right) y =\frac{1}{ x }$
I.F. $=\exp \int\left(\frac{1}{\sin x}+\cot x+\frac{1}{x}\right) d x$
$=\exp \ell n\left(x \tan \frac{x}{2} \sin x\right)$
$=x \tan \frac{x}{2} \times 2 \sin \frac{x}{2} \cos \frac{x}{2}=x(1-\cos x)$
Solution, $y x(1-\cos x)=\int \frac{1}{x} \cdot x(1-\cos x) d x$
$=x-\sin x+c$
$y \left(\frac{\pi}{2}\right)=1-\frac{2}{\pi} \Rightarrow c =0$
$ \therefore y ( x ) =\frac{ x -\sin x }{ x (1-\cos x )} $
$ y = \frac{ x -\left( x -\frac{ x ^{3}}{6} \ldots\right)}{ x \left(1-\left(1-\frac{ x ^{2}}{2} \ldots\right)\right)}$
$=\frac{ x ^{2}}{6} \frac{1}{\frac{ x ^{2}}{2}} $
as $ x \rightarrow 0, y \rightarrow $