Set and
You found an exact ODE, therefore, exists a function such that
and
Integrate the first equation with respect to , after that find the partial derivative with respect to .
I have the problem of (ylny + e^x)dx + (x+e^-y + xlny)dy. I have determined that they are exact. However, I am running into a problem when I am trying to solve the equation. I have done the following and got stuck!
1. partial F with respect to x = ylny + e^x and
2. partial F with respect to y = x + e^-y + xlny
When I integrated #1 with respect to x, I came up with: xylny + e^x + g(y) and then when I integrated #2 with respect to y, I came up with:
xy - e^-y + xylny -x + h(x). From here I am stuck, therefore I think I made a mistake somewhere and I can't seem to find it. Can somebody help?
Hello, graceofayak!
I have the problem of: .
I have determined that they are exact.
I have done the following and got stuck:
.and .
When I integrated #1 with respect to x, I came up with: . Yes!
Then when I integrated #2 with respect to y, I came up with: . . no
Your error is in integrating the third term: .
By parts: .
Then: .
So the integral of #2 is: .