Which Differential Equation Method Do I use?

May 2010
4
0
i need to solve this differential equation (3y^3-xy)dx-(x^2+6xy^2)dy=0

I tried using Bernoulli, but it doesn't work. Also tried separating the variables but i cant seem to separate it. i also tried using the 3 cases for exact equation but to no avail. the 3rd case may work but i don't know how to simplify it so i can integrate it.
 

mr fantastic

MHF Hall of Fame
Dec 2007
16,948
6,768
Zeitgeist
i need to solve this differential equation (3y^3-xy)dx-(x^2+6xy^2)dy=0

I tried using Bernoulli, but it doesn't work. Also tried separating the variables but i cant seem to separate it. i also tried using the 3 cases for exact equation but to no avail. the 3rd case may work but i don't know how to simplify it so i can integrate it.
Are you sure it's not (3y^2 - xy) dx -(x^2 + 6xy^2) dy=0?
 
May 2010
4
0
the problem was hand written, so it may have been a mistake. what if it was squared instead of cube? how do you solve it?
 

Jester

MHF Helper
Dec 2008
2,470
1,255
Conway AR
i need to solve this differential equation (3y^3-xy)dx-(x^2+6xy^2)dy=0

I tried using Bernoulli, but it doesn't work. Also tried separating the variables but i cant seem to separate it. i also tried using the 3 cases for exact equation but to no avail. the 3rd case may work but i don't know how to simplify it so i can integrate it.
If you write the ODE as

\(\displaystyle \frac{dy}{dx} = \frac{3y^3-xy}{x^2 + 6xy^2}\)

multiply both side by 2y

\(\displaystyle 2y\frac{dy}{dx} = \frac{6y^4-2xy^2}{x^2 + 6xy^2}\)

and let \(\displaystyle u = y^2\) then

\(\displaystyle \frac{du}{dx} = \frac{6u^2-2xu}{x^2 + 6xu}\) (this is homogeneous).
 
  • Like
Reactions: lanczlot
May 2010
4
0
wait... how is the differential equation with u homogeneous?
 

Jester

MHF Helper
Dec 2008
2,470
1,255
Conway AR
Divide everything on the rhs by \(\displaystyle x^2\) so

\(\displaystyle
\frac{du}{dx} = \frac{6\dfrac{u^2}{x^2}-2\dfrac{u}{x}}{1 + 6\dfrac{u}{x}}\)

which of the form is \(\displaystyle \frac{du}{dx} = f\left(\frac{u}{x}\right)\).
 

Krizalid

MHF Hall of Honor
Mar 2007
3,656
1,699
Santiago, Chile
the ODE also admits an integrating factor of the form \(\displaystyle u(x,y)=x^my^n.\)