Search any question & find its solution
Question:
Answered & Verified by Expert
If $\frac{d y}{d x}=\frac{y+x \tan \frac{y}{x}}{x}$, then $\sin \frac{y}{x}$ is equal to
Options:
Solution:
1554 Upvotes
Verified Answer
The correct answer is:
$c x$
Given that
This is a homogeneous differential equation.
Put $y=v x$
and
$\frac{d y}{d x}=v+x \frac{d v}{d x}$
From Eq. (i),
$v+x \frac{d v}{d x}=\frac{v x+x \tan \left(\frac{v x}{x}\right)}{x}$
$\Rightarrow \quad x \frac{d v}{d x}=v+\tan v-v$
$\Rightarrow \quad \cot v d v=\frac{d x}{x}$
On integrating both sides, we get
$\begin{aligned}
\Rightarrow & & \log \sin v & =\log x+\log c \\
\Rightarrow & & \sin v & =x c \\
& & \sin \frac{y}{x} & =x c
\end{aligned}$
This is a homogeneous differential equation.
Put $y=v x$
and
$\frac{d y}{d x}=v+x \frac{d v}{d x}$
From Eq. (i),
$v+x \frac{d v}{d x}=\frac{v x+x \tan \left(\frac{v x}{x}\right)}{x}$
$\Rightarrow \quad x \frac{d v}{d x}=v+\tan v-v$
$\Rightarrow \quad \cot v d v=\frac{d x}{x}$
On integrating both sides, we get
$\begin{aligned}
\Rightarrow & & \log \sin v & =\log x+\log c \\
\Rightarrow & & \sin v & =x c \\
& & \sin \frac{y}{x} & =x c
\end{aligned}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.