Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • NCERT
  • solve x\frac{dy}{dx}=y(log y -log x +1)

solve x\frac{dy}{dx}=y(\log y - \log x +1)

Answers (1)

$$ x \frac{d y}{d x}=y(\log y-\log x+1) $$
Using loga-logb =loga/b
$$ \Rightarrow \frac{d y}{d x}=\frac{y}{x}\left(\log \frac{y}{x}+1\right) $$
Put  y=v x 
$$ \Rightarrow \frac{d(v x)}{d x}=\frac{v x}{x}\left(\log \frac{v x}{x}+1\right) $$
Differentiate yx with respect to x using product rule
$$ \Rightarrow \frac{d v}{d x} x+v=v(\log v+1) $$

\Rightarrow \frac{\mathrm{dv}}{\mathrm{dx}} \mathrm{x}+\mathrm{v}=\mathrm{v} \log \mathrm{v}+\mathrm{v}$

$$ \\ \Rightarrow \frac{d v}{d x} x=v \log v \\ \Rightarrow \frac{d v}{v \log v}=\frac{d x}{x} $$
Now Integrate

\Rightarrow \int \frac{\mathrm{dv}}{\text { vlogv }}=\int \frac{\mathrm{dx}}{\mathrm{x}}$

Substitute log v =t
Differentiate with respect to v.

\frac{d v}{v}=d t$
\Rightarrow \int \frac{d t}{t}=\log x+c$
logt= logx  + logc
Resubstitute value of t
log(log v)=log x + logc.
Resubstitute v
$$ \\ \Rightarrow \log \left(\log \frac{y}{x}\right)=\log x+\log c \\ \Rightarrow \log \frac{y}{x}=c x $$
Therefore the solution of the differential equation is
\log \frac{y}{x}=c x

Posted by

infoexpert22

View full answer

Similar Questions

Latest Question