The solution of the differential equation x dy/dx=y-x ≤ft(y/x) is

Correct Answer is : xsin⁡(yx)=Cx ≤ft(y/x) = Cxsin(xy)=C

Correct Answer is : xsin⁡(yx)=Cx ≤ft(y/x) = Cxsin(xy)=C

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CGPET 2013 Solved Paper

Section: Mathematics

Question : 103 of 150

Marks: +1, -0

x \frac{dy}{dx}=y-x \tan\left(\frac{y}{x}\right)

Solution:

Given, differential equation is

x \frac{dy}{dx}=y-x \tan\left(\frac{y}{x}\right)

\Rightarrow \frac{dy}{dx}=\frac{y}{x}-\tan\left(\frac{y}{x}\right)

. . . (i)

Put

y=v x \Rightarrow \frac{dy}{dx}=v+x \frac{dv}{dx}

in Eq. (i), we get

v+x \frac{dv}{dx}=v-\tan v

\Rightarrow x \frac{dv}{dx}=v-\tan v-v=-\tan v

\Rightarrow \frac{dv}{\tan v} = -\frac{dx}{x}

\Rightarrow \frac{\cos v}{\sin v} dv = \frac{-dx}{x}

On integrating both sides, we get

\int \frac{\cos v}{\sin v} dv = -\int \frac{dx}{x}

\Rightarrow \log(\sin v) = -\log x + \log C

\Rightarrow \log(\sin v) + \log x = \log C

\Rightarrow \log(x \sin v) = \log C

\Rightarrow \left( \because \log m + \log n = \log m n \right)

x \sin v = C

On putting

v = \frac{y}{x}

, we get

x \sin \left(\frac{y}{x}\right) = C

which is the required solution.

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