Question

Assertion: $x\sin x\frac{dy}{dx}+(x+x\cos x+\sin x)y$
$=\sin x,y\left(\frac{\pi }{2}\right)=1-\frac{2}{\pi }\Rightarrow \underset{x\to 0}{\lim }y(x)=\frac{1}{3}$
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. $=\exp \int \left(\frac{1}{\sin x}+\cot x+\frac{1}{x}\right)dx$
$=\exp \ell n\left(x\tan \frac{x}{2}\sin x\right)$
$=x\tan \frac{x}{2}\times 2\sin \frac{x}{2}\cos \frac{x}{2}=x(1-\cos x)$
Solution, $yx(1-\cos x)=\int \frac{1}{x}\cdot x(1-\cos x)dx$
$=x-\sin x+c$
$y\left(\frac{\pi }{2}\right)=1-\frac{2}{\pi }\Rightarrow c=0$
$\therefore y(x)=\frac{x-\sin x}{x(1-\cos x)}$
$y=\frac{x-\left(x-\frac{x^3}{6}\dots \right)}{x\left(1-\left(1-\frac{x^2}{2}\dots \right)\right)}$
$=\frac{x^2}{6}\frac{1}{\frac{x^2}{2}}$
as $x\to 0,y\to$