Question

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

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

Answer 1

Answer: (A)

Given differential equation can be rewritten as
$\frac{dy}{dx}+2y\cot x=cosecx$
It is a linear differential equation of the form
$\frac{dY}{dx}+P_Y=Q$
Here, $P=2\cot x$ and $Q=cosecx$
$\therefore IF=e^{\int Pdx}=e^{2\int \cot xdx}$
$=e^{2\log \sin x}=e^{\log {\sin }^{2}x}={\sin }^{2}x$