[ xy dx-(x^{2}+3y^{2})dy=0

avissidep 2020-10-26 Answered

 xydx(x2+3y2)dy=0

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Expert Answer

yunitsiL
Answered 2020-10-27 Author has 108 answers
Write this equation as
 (x2+3y2)dy=xydxdydx=xyx2+3y2=yx1+3y2x2
(In the last equality, we divided the numerator and denominator by x2 .) Therefore, the equation becomes is
dydx=yx1+3y2x2
(1) which is of the form
dydx=h(yx)
Thus, we will use the substitution
y=uxdydx=dudxx+u (2)
Plugging (2) into (1), we get that,
dudxx+u=u1+3u2xdudx=u1+3u2u=uu(1+3u2)1+3u2=3u21+3u2 Therefore,
xdudx=3u21+3u21+3u23u3du=dxx
Integrating both sides:
1+3u23u3du=dxx
Now,
dxx=ln|x|+C2=ln1|x|+C2
where C2 is some constant. (We have also used that nlny=lnyn)
On the other hand,
1+3u23u3du=13duu3+duu
=13u3du+ln|u|
=16u2+ln|u|+C1
=16u2+ln|u|+C1
Here C1 is some constant
Using (3) we now get that
16u2+ln|u|+C1=ln1|x|+C2
Defining C=C2C1, we get
16u2+ln|u|=ln1|x|+C
Finally, recall that u=yx to write the final solution (in the implicit form)
ln|yx|x26y2=ln1|x|+C
Answer ln|yx|x26y2=ln1|x|+C
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