Question

An integrating factor of the differential equation $\sin x\frac{dy}{dx}+2y\cos x=1$ is

• A. ${\sin }^{2}x$
• B. $\frac{2}{\sin x}$
• C. $\log |\sin x|$
• D. $\frac{1}{{\sin }^{2}x}$
• E. $2\sin x$

Answer 1

Answer: (A)

${\sin }^{2}x$

$\sin x\frac{dy}{dx}+2y\cos x=1$

$\Rightarrow \frac{dy}{dx}+2y\cot x=\csc x$

Here, $P=2\cot x,Q=\csc x$

Integrating factor $I=e^{\int Pdx}$

$=e^{\int 2\cot xdx}=e^{2\int \frac{\cos x}{\sin x}dx}$

$=e^{2\ln \sin x}$ Integration of $\int \cot x=\ln |\sin x|+C$

$={\sin }^{2}x$