How do you solve the ODE dy/dx=1/(y+x)?
Just had an exam and I got stuck with this DE
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.
Please help me solve this differential equation!
- az_lenderLv 76 years agoBest Answer
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.
- JeffLv 76 years ago
Hint: Let u(x) = y(x) + x