Question

The integrating factor of the differential equation $\frac{dy}{dx}=ytanx-y^{2}secx$ is

[MP PET 1995; Pb. CET 2002]

• A. tan x
• B. secx
• C. -sec x
• D. cot x

Answer 1

The correct option is B

secx

The differential equation
is $\frac{dy}{dx}-ytanx=-y^{2}secxI.F.=e^{-\int tanxdx}$
This is Bernoulli's equation i.e. reducible to
linear equation.
Dividing the equation by $y^2,$ we get
$\frac{1}{y^2}\frac{dy}{dx}-\frac{1}{y}tanx=-secx............\left(i\right)$
Put $\frac{1}{y}=y\Rightarrow -\frac{1}{y^2}\frac{dy}{dx}=\frac{dY}{dx}$
Equation (i) reduces to $-\frac{dy}{dx}=-ytanx=-secx\Rightarrow \frac{dY}{dx}+Ytanx=secx,$ Which is a linear equation
Hence $I.F.=e^{-\int tanxdx}=secx.$