Question

Need help making an nonexact DE exact!

(2xy^3+y^4)dx+(xy^3-2)dy

Answer 1

are you looking for an integrating factor?

Answer 2

I have tried \[{M_y-N_x}/N \] and \[ {N_x-M_y}/M\] but neither seem to be function of x or y alone (respectively). Not sure where to go as my math department is useless and all I have is the textbook to study from, which doesn't seem to help either

Answer 3

yes I think I am looking for an integrating factor. I am looking to solve it however that may be

Answer 4

i guess if you have proved the integrating factor is not a factor of x or y alone,

you cab expect the integrating factor to be a function of x and y

Answer 5

well I am not sure that is explained in my book. How do I use it then?

Answer 6

what symbol do you normally use fot the integrating factor

Answer 7

\[\mu\]

Answer 8

\[\mu=\mu(x,y)=x^my^n\]
\[\qquad\qquad\qquad\frac{\partial \mu(x,y)M}{\partial y}= \frac{\partial \mu(x,y)N}{\partial x}\]
\[\mu_y(x,y)M+\mu(x,y)\frac{\partial M}{\partial y}=\mu_x(x,y)N+\mu(x,y)\frac{\partial N}{\partial x}\]

Answer 9

does this make sense?

Answer 10

not really. is this still and ODE? seems like a PDE to me

Answer 11

it is still a ODE,
take the partial derivatives and substitute

Answer 12

\[\mu=x^my^n\]\[\mu_x=mx^{m-1}y^n\qquad\qquad \mu_y=nx^{m}y^{n-1}\]

Answer 13

\[M=2xy^3+y^4\]\[M_y=6xy^2+4y^3\qquad\qquad M_x=2y^3\]

Answer 14

if you want another text on first order ODE's
,
it dosent really go into all the details when \(\mu=\mu(x,y)\)
but it kinda tell you what to do

Answer 15

oh wow, thanks a lot for sharing that. I will give it a read and try again here