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

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\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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