## anonymous 5 years ago How do I solve the following differential equation? Thanks. dy/dx=(1-x-y)/(x+y)

1. anonymous

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

2. anonymous

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.

3. anonymous

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$\frac{\partial \psi }{\partial y}=N$

4. anonymous

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

5. anonymous

Okay, set u=x+y

6. anonymous

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

7. anonymous

8. anonymous

yes yes! that's the stuff thank you

9. anonymous

okay