Been stuck on this for quite a bit. The goal is to find the solution to the initial value problem:
$$(e^{x+y}+e^y)dx + (e^{x+y}+e^x)dy = 0, where...$$ $$y(0)=0$$
So the first thing I did was check for exactness - the equation is in the form $M(x,y)dx + N(x,y)dy =0$, so taking the partial derivative with respect to y of #M(x,y)# and the same with respect to x of $N(x,y)$ doesn't change either expression. They're not equal, so the equation is inexact. Okay, cool.
Next thing I did was look for an integrating factor - we were taught to check two expressions, $(Nx-My)/N$, and $(My-Nx)/M$. I did that, and both of the resulting expressions were a function of both x and y. In this case, I'm not sure what to do - our textbook only covers the case where one or the other is either a function of x or y only.
Have I made a mistake somewhere? How can I move forwards here? I feel like I need to utilize the initial condition somehow but I haven't been able to figure out how.
Thanks for any advice you can give!
It is separable.
$(e^{x+y}+e^y)dx + (e^{x+y}+e^x)dy = 0$
Dividing by $e^{x+y}$,
$(1 + e^{-x})dx + (1 + e^{-y})dy = 0$
$(1 + e^{-x})dx = -(1 + e^{-y})dy$