GATE (TF) Textile 2017 Question Paper Solution | GATE/2017/TF/28

Question 28 (Textile Engineering & Fibre Science)

If a solution curve of the differential equation $x\frac{dy}{dx}=y+2x^3$ passes through the point (1,0), then this curve also passes through the point

(A)	(-1,0)
(B)	(0,-1)
(C)	(2,10)
(D)	(-2,6)

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$x\frac{dy}{dx}=y+2x^3$
$\frac{dy}{dx}-\frac{1}{x}\times y=2x^2$
Here, $P=-\frac{1}{x} and Q=2x^2$
Here, $I.F.=e^{-\int \frac{1}{x} dx}$
$I.F.=e^{-log x}$
$I.F.=e^{log\frac{1}{x}}$
$I.F=\frac{1}{x}$
Then solution-
$y(I.F.)=\int Q \times I.F.+c$
$y(\frac{1}{x})=\int 2x^2 \times \frac{1}{x}+c$
$y(\frac{1}{x})=\int 2x+c$
$y(\frac{1}{x})=\frac{2x^2}{2}+c$
$y(\frac{1}{x})=x^2+c$
$y=x^3+cx$
Given ,this differential equation passses through point (1,0)
$0=1^3+c \times 1$
c=-1
Now, $y=x^3+(-1)x$
$y=x^3-x$
So, the equation also passes through the Point (-1,0).(Ans)