## nubeer 3 years ago Calc 3... how do we find the integration factor of equation which is not exact.. (i want to know for in which both x and y is used as integration factor)

1. UnkleRhaukus

do you have an example we can work through /

2. nubeer

hmm no i don't have any example .. we were asked to look for it.. i just want to know any general method.. any link or inofrmation would appreciate it.. its basically part of diffrential equations.

3. UnkleRhaukus

$\left(3x+\frac 6y\right)\text dx+\left(\frac {x^2}{y}+\frac{3y}{x}\right)\text dy =0$

4. UnkleRhaukus

$\frac{\partial M}{\partial y}=\frac {-6}{y^{2}}\qquad\qquad \frac{\partial N}{\partial x}=\frac{2x}{y}-\frac{3y}{x^2}$$\qquad\qquad\frac{\partial M}{\partial y}\neq \frac{\partial N}{\partial x}$

5. nubeer

ok..

6. UnkleRhaukus

$R=R(x,y)=x^my^n$

7. nubeer

ok this is soemthing new.. is this the general form of the solution?

8. UnkleRhaukus

im using R for the integrating factor

9. nubeer

ok..

10. UnkleRhaukus

$\frac{\partial R(x,y)M}{\partial y}\quad= \frac{\partial R(x,y)N}{\partial x}$ using the product rule $R_y(x,y)M+R(x,y)\frac{\partial M}{\partial y}=R_x(x,y)N+R(x,y)\frac{\partial N}{\partial x}$

11. nubeer

ok i get this much

12. UnkleRhaukus

$\small R_y(x,y)\left(3x+\frac 6y\right)+R(x,y)\left(\frac {-6}{y^{2}}\right)=R_x(x,y)\left(\frac {x^2}{y}+\frac{3y}{x}\right)+R(x,y)\left(\frac{2x}{y}-\frac{3y}{x^2}\right)$ $\small nx^my^{n-1}\left(3x+\frac 6y\right)+x^my^n\left(\frac {-6}{y^{2}}-\frac{2x}{y}+\frac{3y}{x^2}\right)-mx^{m-1}y^n\left(\frac {x^2}{y}+\frac{3y}{x}\right)=0$

13. UnkleRhaukus

.... somehow find n and m ,

14. nubeer

how? any particular method?

15. UnkleRhaukus

something to do with the matching indices

16. nubeer

hmm still dont get the last part..

17. UnkleRhaukus

me either

18. nubeer

hmm ok its fine but thanks for the help bro.. :) will look on internet

19. UnkleRhaukus

if you find a really good sight can you tell me please

20. nubeer

sure i will let u know..

21. nubeer

http://www.cliffsnotes.com/study_guide/Integrating-Factors.topicArticleId-19736,articleId-19711.html @UnkleRhaukus this is a good one but it dont tell for the both x and y .. will post it if i find it