GATE (TF) Textile 2014 Question Paper Solution | GATE/2014/TF/28

Question 28 (Textile Engineering & Fibre Science)

The integrating factor for solving the differential equation $\frac{dy}{dx}\cos(x)+y\sin(x)=2\cos ^3(x)$ is

(A)	$\sin(x)$
(B)	$\cos(x)$
(C)	$\tan(x)$
(D)	$\sec(x)$

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Given-
$\frac{dy}{dx}\cos(x)+y\sin(x)=2\cos ^3(x)$
$\frac{dy}{dx}+(tanx) y=2 cos^2(x)$
Compare with the differential equation-
P=tanx and Q= $2cos^2 (x)$
Integrating factor(I.F.)= $e^{\int P dx}$
Integrating factor(I.F.)= $e^{\int tanx dx}$
Integrating factor(I.F.)= $e^{\int \frac{sinx}{cosx} dx}$
Let, cosx=t
differentiate this-
-sinx dx=dt
Integrating factor(I.F.)= $e^{\int \frac{-1}{t} dt$
Integrating factor(I.F.)= $e^{-log t}$
Integrating factor(I.F.)= $e^{log\frac{1}{t}}$
Integrating factor(I.F.)= $\frac{1}{t}$
Integrating factor(I.F.)= $\frac{1}{cosx}$
Integrating factor(I.F.)=sec x (Ans)
Option D is correct.

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