Differential Equations Question 227
Question: The integrating factor of the differential equation $ \frac{dy}{dx}=y\tan x-y^{2}\sec x, $ is
[MP PET 1995; Pb. CET 2002]
Options:
A) $ \tan x $
B) $ \sec x $
C) $ -\sec x $
D) $ \cot x $
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Answer:
Correct Answer: B
Solution:
The differential equation is $ \frac{dy}{dx}-y\tan x=-y^{2}\sec x $
I.F. $ ={e^{-\int _{{}}^{{}}{\tan xdx}}} $
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}\tan x=-\sec x $
…..(i) Put $ \frac{1}{y}=Y $
Therefore $ -\frac{1}{y^{2}}\frac{dy}{dx}=\frac{dY}{dx} $
Equation (i) reduces to $ -\frac{dY}{dx}-Y\tan x=-\sec x $
Therefore $ \frac{dY}{dx}+Y\tan x=\sec x $ ,which is a linear equation Hence I.F. $ ={e^{\int _{{}}^{{}}{\tan x}dx}}=\sec x $ .
BETA
