DE#3

Dec 2015
1,085
169
Earth
\(\displaystyle dx+(e^y -x)dy=0\).
 

topsquark

Math Team
May 2013
2,533
1,052
The Astral plane
Just out of curiousity, where are you getting these DEqs from?

-Dan
 
Dec 2015
1,085
169
Earth
Just out of curiousity, where are you getting these DEqs from?

-Dan
from a very old book , it is an exact DE.
 
  • Like
Reactions: topsquark

topsquark

Math Team
May 2013
2,533
1,052
The Astral plane
Well, if it's exact then you have
\(\displaystyle 1 + \left ( e^y - x \right ) \dfrac{dy}{dx} = 0\)

and you have the general form
\(\displaystyle M(x, y) + N(x, y) \dfrac{dy}{dx} = 0\)

then what are M(x, y) and N(x, y) and how do you use them?

Let us know what you've got and we'll go from there. The solution method isn't particularly hard but it does require some intuition.

-Dan
 
  • Like
Reactions: idontknow

skipjack

Forum Staff
Dec 2006
21,482
2,472
The equation (which isn't exact) implies \(\displaystyle e^{-y}\frac{dx}{dy} - xe^{-y} = -1\).
Integrating with respect to y gives $e^{-y}x = -y + \text{C}$, where $\text{C}$ is a constant,
so $x = (\text{C} - y)e^y$.
 

topsquark

Math Team
May 2013
2,533
1,052
The Astral plane
The equation (which isn't exact) implies \(\displaystyle e^{-y}\frac{dx}{dy} - xe^{-y} = -1\).
Integrating with respect to y gives $e^{-y}x = -y + \text{C}$, where $\text{C}$ is a constant,
so $x = (\text{C} - y)e^y$.
Nice!

-Dan