Solution of the differential equationxdy/dx=y-x≤ft(y/x) will be

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

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

Test Index

CGPET 2018 Solved Paper

Section: Mathematics

Question : 102 of 150

Marks: +1, -0

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

will be

Solution:

Given differential equation

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

⇒

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

........(i)

which is homogeneous differential equation.

Now, put

\frac{y}{x}=v

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

.........(ii)

From Eqs. (i) and (ii), we get

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

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

\Rightarrow \cot v\, dv=-\frac{1}{x}dx

On integrating both the sides, we get

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

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

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

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

Go to Question:

Prev Question

Next Question