# How do you solve the ODE dy/dx=1/(y+x)?

Just had an exam and I got stuck with this DE

dy/dx=1/(y+x)

or

-dx+(x+y)dy=0

The homogeneous test doesn't work, I can't normalize it, I cant separate it. I tried doing almost exact formula (My-Mx)/M or N but it gave me an integrating factor of e^x or e^-y, which didn't work.

Relevance
• 6 years ago

If u = x+y, then du/dx = 1 + dy/dx, and

du/dx - 1 = 1/u

du/dx = 1 + 1/u = (u+1)/u

u du/(u+1) = dx

Now it's separated; to integrate the LHS, let v=u+1, so

(1 - 1/v) dv = dx

v - ln|v| = x + C1

u - ln|u+1| = x + C2

x + y - ln|x+y+1| = x + C2

y = ln|x+y+1| + C2

ce^y = x + y + 1.

• Jeff
Lv 7
6 years ago

Hint: Let u(x) = y(x) + x