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Mathematics
OpenStudy (unklerhaukus):

Solve the differential equation \[{(3xy+y^2)+(x^2+xy)y'=0}\] Using the integrating factor \[R(x,y)=\frac{1}{xy(2x+y)}\]

OpenStudy (unklerhaukus):

\[\text{We require for exactness that}\]\[\frac{\partial RM}{\partial y}=\frac{\partial RN}{\partial x}\]\[R(x,y)=\frac{1}{xy(2x+y)}\]\[{R(x,y)(3xy+y^2)+R(x,y)(x^2+xy)y'=0}\]

OpenStudy (unklerhaukus):

\[{\frac{(3x+y)}{x(2x+y)}+\frac{(x+y)}{y(2x+y)}y'=0}\]\[RM=\frac{(3x+y)}{x(2x+y)}\]\[\frac{\partial RM}{\partial y}=-\frac{1}{(2x+y)^2}\]\[RN=\frac{(x+y)}{y(2x+y)}\]\[\frac{\partial RN}{\partial x}=-\frac{1}{(2x+y)^2}\]\[\frac{\partial RM}{\partial y}=\frac{\partial RN}{\partial x}\]

OpenStudy (unklerhaukus):

\[\frac{\partial f(x,y)}{\partial x}=R(x,y)M\qquad\qquad\frac{\partial f(x,y)}{\partial y}=R(x,y)N\]\[f(x,y)=\int R(x,y)M\text dx\]\[=\int {\frac{(3x+y)}{x(2x+y)}\text dx+g(y)}\]

OpenStudy (unklerhaukus):

\[=\frac 12 \log(2x+y)+\log(2x)+g(y)\]

OpenStudy (experimentx):

alternatively ... looks like homogeneous eqaution

OpenStudy (unklerhaukus):

im am trying to solve it using the integrating factor give in the question, this is as far as i have got if you can solve if another way can you tell me the result. so i can check

OpenStudy (experimentx):

lol ... i like the wolfram solution best!!

OpenStudy (unklerhaukus):

what did you input into wolfram ,?

OpenStudy (unklerhaukus):

Interpreting "dfferentiate" as "differentiate" lolz

OpenStudy (unklerhaukus):

that is how i got there too

OpenStudy (unklerhaukus):

shouldn't the thing with the logs equal RN? something ins't right

OpenStudy (unklerhaukus):

isn't*

OpenStudy (experimentx):

Looks like I forgot something!!

OpenStudy (experimentx):

seems all right \[ g(y) = \int \frac{x+y}{y(2x+y)} dy\]

OpenStudy (experimentx):

there isn't any function of x http://www.wolframalpha.com/input/?i=integrate+%28x%2By%29%2F%28y%282x%2By%29%29+with+respect+to+x

OpenStudy (experimentx):

wolfram is giving complicated results http://www.wolframalpha.com/input/?i=%283xy%2By^2%29dx%2B%28x^2%2Bxy%29+dy%3D0 try viewing steps!!

OpenStudy (unklerhaukus):

yep

OpenStudy (unklerhaukus):

well i dont know what the correct answer is

OpenStudy (unklerhaukus):

i dont know i might try another i have been working on today

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