Question

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

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

alternatively ... looks like homogeneous eqaution

Answer 6

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

Answer 7

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

Answer 8

what did you input into wolfram ,?

Answer 9

currently checking your partial derivatives!!
http://www.wolframalpha.com/input/?i=dfferentiate+%28x%2By%29%2F%28y%282x%2By%29%29+with+respect+to+x
http://www.wolframalpha.com/input/?i=dfferentiate+%28x%2By%29%2F%28y%282x%2By%29%29+with+respect+to+x

Answer 10

Interpreting "dfferentiate" as "differentiate" lolz

Answer 11

seems like you are right
http://www.wolframalpha.com/input/?i=integrate+%283x%2Ba%29%2F%28x%282x%2Ba%29%29+with+respect+to+x

Answer 12

that is how i got there too

Answer 13

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

Answer 14

isn't*

Answer 15

Looks like I forgot something!!

Answer 16

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

Answer 17

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

Answer 18

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!!

Answer 19

yep

Answer 20

well i dont know what the correct answer is

Answer 21

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

Answer 22

I don't now why i am getting this
http://www.wolframalpha.com/input/?i=%28integrate+%283x%2By%29%2F%28x%282x%2By%29%29+with+respect+to+x%29+%2B+%28integrate+%28x%2By%29%2F%28y%282x%2By%29%29+with+respect+to+y%29