Question

How do I solve the following differential equation? Thanks.
dy/dx=(1-x-y)/(x+y)

Answer 1

It should be some type of u substitution, we're working on different form of separation of variables.

Answer 2

This is an exact differential equation. You can write it in the form\[(x+y-1)+(x+y)\frac{dy}{dx}=0\]Identify,\[M=x+y-1\]and\[N=x+y\]Then,\[\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}(=1)\]which shows this is exact.

Knowing this, you can move through with solving it.

Answer 3

In this method, you're looking for a function\[\psi(x,y)=c\]where c is some constant. The proof of the method shows this function should have a form such that\[\frac{\partial \psi}{\partial x}=M\]and\[\frac{\partial \psi }{\partial y}=N\]

Answer 4

Okay this looks good. Thank you, but I think I have to do this using I substitution.

Answer 5

Okay, set u=x+y

Answer 6

then du=dx and you have\[\frac{dy}{du}=\frac{1-u}{u}=\frac{1}{u}-1 \rightarrow y=\log u -u=\log (x+y) -(x+y) +c\]

Answer 7

add a constant

Answer 8

yes yes! that's the stuff
thank you

Answer 9

okay