Email
Chat with tutor
logo

Ask Questions, Get Answers

X
 
Answer
Comment
Share
Q)

Solve: $(3xy+y^2)dx+(x^2+xy)dy=0 \qquad y(1)=1$

$(a)\;x^2(2x+y)=3 \\ (b)\;y^2(2x+y)=3 \\ (c)\;x^2y(2x+y)=3 \\ (d)\;x^2y(2x+3)=9 $

1 Answer

Comment
A)
$\large\frac{dy}{dx}=-\large\frac{3xy+y^2}{x^2+xy}$
$y=vx$
$v+xv'= -\large\frac{3v+v^2}{1+v}$
$xv'= -[-v+\large\frac{3v+v^2}{1+v}]$
$\qquad =\large\frac{-2v (2+v)}{1+v}$
Separating variables ,
$\large\frac{(1+v)dv}{v(2+v)}=\large\frac{-2}{x} $$dx$
$\large\frac{1}{2} \bigg[\large\frac{1}{v}+\frac{1}{2+v}\bigg]dv=-\large\frac{-2}{x}$$dx$
Integrate $\log (v+2+v))+4 \log x =2 \log c$
$v (2+v)x^4=c^2$
$v=\large\frac{y}{x}$
$x^2y (2x+y)=c^2=k$
=> $y(1)=1$
$x^2y (2x+y)=3$
Hence c is the correct answer.
Help Clay6 to be free
Clay6 needs your help to survive. We have roughly 7 lakh students visiting us monthly. We want to keep our services free and improve with prompt help and advanced solutions by adding more teachers and infrastructure.

A small donation from you will help us reach that goal faster. Talk to your parents, teachers and school and spread the word about clay6. You can pay online or send a cheque.

Thanks for your support.
Continue
Please choose your payment mode to continue
Home Ask Homework Questions
Your payment for is successful.
Continue
Clay6 tutors use Telegram* chat app to help students with their questions and doubts.
Do you have the Telegram chat app installed?
Already installed Install now
*Telegram is a chat app like WhatsApp / Facebook Messenger / Skype.
...