anonymous
  • anonymous
How do I solve the following differential equation? Thanks. dy/dx=(1-x-y)/(x+y)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
It should be some type of u substitution, we're working on different form of separation of variables.
anonymous
  • 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.
anonymous
  • 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\[\frac{\partial \psi }{\partial y}=N\]

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anonymous
  • anonymous
Okay this looks good. Thank you, but I think I have to do this using I substitution.
anonymous
  • anonymous
Okay, set u=x+y
anonymous
  • anonymous
then du=dx and you have\[\frac{dy}{du}=\frac{1-u}{u}=\frac{1}{u}-1 \rightarrow y=\log u -u=\log (x+y) -(x+y) +c\]
anonymous
  • anonymous
add a constant
anonymous
  • anonymous
yes yes! that's the stuff thank you
anonymous
  • anonymous
okay

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