The solution of the differential equation x \;dy/dx=y+x \;y/x {is}

Correct Answer is : sin⁡ yx=Cx \;\;y/x=C xsinxy=Cx

Correct Answer is : sin⁡ yx=Cx \;\;y/x=C xsinxy=Cx

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CGPET 2012 Solved Paper

Section: Mathematics

Question : 111 of 150

Marks: +1, -0

x \;\frac{dy}{dx}=y+x \tan\;\frac{y}{x} \text{is}

Solution:

\;x \;\frac{dy}{dx}=y+x \tan\;\frac{y}{x}

\;\Rightarrow \;\; \;\frac{dy}{dx}=\;\frac{y+x \tan\frac{y}{x}}{x}

It is a homogeneous differential equation.

Put

y=v x

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

\Rightarrow\;v+x \;\frac{dv}{dx}\;=\;\frac{v x+x \tan\frac{v x}{x}}{x}

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

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

\Rightarrow\;\tan v\;=\;\frac{dx}{x}

\Rightarrow \cot v d v\;=\;\frac{dx}{x}

On integrating both sides, we get

\;\log \sin \;v\;=\log x+\log C

\Rightarrow\;\log \sin \;v\;=\log x C

\Rightarrow \sin \;v\;=x C

\therefore \; \sin \;\;\frac{y}{x}\;=C x \;\; \left( \because \;\text{ Put }\; v=\frac{y}{x}\right)

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