A) \[\tan x\]
B) \[\sec x\]
C) \[-\sec x\]
D) \[\cot x\]
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 \[\frac{1}{y}=Y\] Þ \[-\frac{1}{{{y}^{2}}}\frac{dy}{dx}=\frac{dY}{dx}\] Equation (i) reduces to \[-\frac{dY}{dx}-Y\tan x=-\sec x\] Þ \[\frac{dY}{dx}+Y\tan x=\sec x\],which is a linear equation Hence I.F. \[={{e}^{\int_{{}}^{{}}{\tan x}\,dx}}=\sec x\] .
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