• Support PF! Buy your school textbooks, materials and every day products Here!

Exact Equation with Integrating Factor Help

  • #1

Homework Statement



Multiply the given equation by the given integrating factor and solve the exact equation.



Homework Equations



ydx+(2x-yey)dy=0, [tex]\mu[/tex](x,y)=y.

The Attempt at a Solution



M=y2, N=2xy-y2ey

Integrating N=[tex]\Psi[/tex]y WRT x I get

xy2-((1/3)y3ey + y2ey)+h(x)=[tex]\Psi[/tex](x,y)

Differentiating [tex]\Psi[/tex](x,y) WRT x, I get

[tex]\Psi[/tex]x=y2+h'(x)

Thus h'(x)=0, and

[tex]\Psi[/tex](x,y)=xy2-((1/3)y3ey + y2ey)

The correct answer is xy2-(y2-2y+2)ey=c...

What am I doing wrong? When I solve for psi in the opposite way I get the same wrong answer from before...

Thanks in advance!
 

Answers and Replies

  • #2
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,535
751

Homework Statement



Multiply the given equation by the given integrating factor and solve the exact equation.



Homework Equations



ydx+(2x-yey)dy=0, [tex]\mu[/tex](x,y)=y.

The Attempt at a Solution



M=y2, N=2xy-y2ey

Integrating N=[tex]\Psi[/tex]y WRT x I get
But you are looking for a function ψ such that ψx = M and ψy = N.

So to find ψ by integrating ψy = N, you need to integrate both sides with respect to y, not x.
 
  • #3
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,305
998
This is my take on this.

Multiplying by the integrating factor gives: y2dx+(2xy-y2ey)dy=0

y2dx/dy+2xy=y2ey

(d/dy)(xy2)=y2ey

Integrate both sides WRT y.

[tex]\int \frac{d}{dy}(xy^2)\ dy=\int y^2\,e^y\, dy[/tex]

So, to do it your way,

Integrate M=y2 WRT x → Ψ(x,y)=xy2+h(y)

 → N=Ψy(x,y) → 2xy+h'(y)=2xy-y2ey

 h'(y) = -y2ey

Integrate this & plug it back into Ψ.
 
Last edited:
  • #4
Great, thanks for the help!
 

Related Threads on Exact Equation with Integrating Factor Help

Replies
15
Views
4K
Replies
2
Views
4K
Replies
4
Views
5K
Replies
2
Views
1K
  • Last Post
Replies
5
Views
5K
Replies
2
Views
6K
Replies
4
Views
566
Replies
0
Views
887
Top