first order ode question non linear

[PapayaC]

FInd the general solution of the following ode.

Question: xe^(x+y)=y*dy/dx

What I could do so far:

e^(x+y)=y dy/ x dx
x+y=ln y/x * dy/dx
x+ln x dx = ln y - y dy
(x^2)/2 + x lnx -x = y lny - y
(x^2)/2 + x lnx -x = y(ln y)
So my problem is that I don´t know if i can take ln on both sides and ignore the y´ (step 2)
my next problem is that i don´t know how to make y the subject here, so that my answer looks cleaner.
If anyone can help that would be great Thanks in advance.

 

[HallsofIvy]

You have an error in your second line. Taking the logarithm of both sides of \(\displaystyle xe^{x+y}= y\frac{dy}{dx}\) you get \(\displaystyle ln(x)+ x+ y= ln(y)+ ln\left(\frac{dy}{dx}\right)\). That is, you have to take the logarithm of the derivative as well.

If you are going to complain that you don't know what a logarithm of a derivative means, I would say "Yes, so don't do that!"

Instead, use the fact that \(\displaystyle e^{x+y}= e^xe^y\) to "separate the variables"
\(\displaystyle xe^{x+ y}= xe^xe^y= y\frac{dy}{dx}\) so
\(\displaystyle xe^x dx= ye^{-y}dy\)

Now, integrate both sides (using "integration by parts").