Question

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)

Answer 1

do you have an example we can work through /

Answer 2

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.

Answer 3

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

Answer 4

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

Answer 5

ok..

Answer 6

\[R=R(x,y)=x^my^n\]

Answer 7

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

Answer 8

im using R for the integrating factor

Answer 9

ok..

Answer 10

\[\frac{\partial R(x,y)M}{\partial y}\quad= \frac{\partial R(x,y)N}{\partial x}\]

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

Answer 11

ok i get this much

Answer 12

\[\small R_y(x,y)\left(3x+\frac 6y\right)+R(x,y)\left(\frac {-6}{y^{2}}\right)=R_x(x,y)\left(\frac {x^2}{y}+\frac{3y}{x}\right)+R(x,y)\left(\frac{2x}{y}-\frac{3y}{x^2}\right)\]

\[\small nx^my^{n-1}\left(3x+\frac 6y\right)+x^my^n\left(\frac {-6}{y^{2}}-\frac{2x}{y}+\frac{3y}{x^2}\right)-mx^{m-1}y^n\left(\frac {x^2}{y}+\frac{3y}{x}\right)=0\]

Answer 13

.... somehow find n and m ,

Answer 14

how? any particular method?

Answer 15

something to do with the matching indices

Answer 16

hmm still dont get the last part..

Answer 17

me either

Answer 18

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

Answer 19

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

Answer 20

sure i will let u know..

Answer 21

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