From Elementary Differential Equation by Boyce and Diprima
Chapter 2 Miscellaneous Problems #11
(x^2+y)dx + (x+e^x)dy = 0
multiplying an integrating factor to make the DE exact:
1. du/dx = u(My - Nx)/ N
2. du/dx = u(Nx-My)/ M
The Attempt at a Solution
First try: I guessed this can be changed into exact DE so, I tried with the two above equation:
equation 1 gave me:
du/u = e^x/(x+e^x)
I don't know how to solve this...
then equation 2 gave me:
u = e^((e^x)*ln(x^2+y))
I am not sure if multiply this integrating factor to the original DE will make it exact...
Second try: I manipulated the given DE and changed it to a linear form:
dy/dx = -(x^2+y)/(x+e^x)
dy/dx + 1/(x+e^x) * y = (-x^2)/(x+e^x)
and I found integrating factor to be:
I = e^∫1/(x+e^x) dx
which I am unable to solve...