Question

T(My-Nx)=NTx-MTy
find T? T(x,y),M(x,y) and N(x,y).
My represents here partial derivative of M w.r.t y, and similarly others.

Answer 1

\[T(M_y-N_x)=NT_x-MT_y\] ?

Answer 2

yes

Answer 3

any idea??

Answer 4

I tried to solve by T=X(x) Y(y)
but did'nt work

Answer 5

oh i remember that way
thats my favorite way because its easiest
i was just looking in my partial differential equation book because i do not remember how

Answer 6

OK

Answer 7

but what did you do with N and M?

Answer 8

I leave them. assume them as function of x and y

Answer 9

help me think
i don't know if i'm going anywhere but i did this:
Let T=X(x)Y(x), M=G(x)F(y), N=H(x)L(y)
...
then I eventually got
XG(YF)'=LY(HX)'

but I don't know if this is gonna take me anywhere

Answer 10

you are assuming T as a function of x,
in that case T=exp(-MTy/My-Nx)

Answer 11

we can put all of our functions of y on one side and all of our functions of x on the other side
(YF)'/(LY)=(HX)'/(XG)
...

Answer 12

M and N will be given so we don't need to decompose them

Answer 13

we just want to find out T

Answer 14

And I want to solve for case when T(x,y)

Answer 15

jamal i give up for now i'm sorry

Answer 16

I found one T=yf(x)
then T=yexp((y(My-Nx)+M)/N)
but this is not the most genral case

Answer 17

And I found this solution by discussing you thank you.

Answer 18

i need to review my differential/partial differential equations :(

Answer 19

you're so sweet jamal
that makes me feel better

Answer 20

:)