Question

Find the integrating factor of the differential equation \( \sin x \frac{dy}{dx} - y \cos x = 1 \):

• A. \(\sin x\)
• B. \(\cos x\)
• C. \(\sec x\)
• D. \(\csc x\)

Hint:

In linear differential equations, the integrating factor is essential for solving the equation and is found by exponentiating the integral of the coefficient of \(y\) in the rearranged form.

Answer 1

The Correct Option is D

The differential equation given is: \[ \sin x \frac{dy}{dx} - y \cos x = 1 \] This can be rewritten in the standard linear form: \[ \frac{dy}{dx} + P(x)y = Q(x) \] where \( P(x) = -\frac{\cos x}{\sin x} \). The integrating factor (I.F.) is given by: \[ I.F. = e^{\int P(x) \, dx} = e^{\int -\cot x \, dx} \] Since the integral of \(-\cot x\) is \(\log |\sin x|\), the I.F. becomes: \[ e^{\log |\sin x|} = |\sin x| \] However, the absolute value is not necessary as we consider only the non-negative part for the integrating factor in this context, hence: \[ I.F. = \csc x \]