## cwrw238 Group Title Whats the integrating factor for the following FODE - I've worked it out but im not sure if its right. (x+1)f' - xy = x one year ago one year ago

1. cwrw238 Group Title

that should be (x+1) y' - xy = x

2. cwrw238 Group Title

first divide through by (x+1) right?

3. Spacelimbus Group Title

So far I end up at $\large \mu(x) = (x+1)e^{-x}$ In case you got the same, otherwise I would have to take a look again

4. cwrw238 Group Title

yes thats what i got

5. Spacelimbus Group Title

alright, in this case we just need to confirm now if it works out, for the LHS it should be possible to be written in the form of a product rule.

6. cwrw238 Group Title

right so its (1+x)e^(-x) y' - x e^(-x) y = x e^(-x) ok?

7. cwrw238 Group Title

i must admit i'm struggling a bit with these

8. Spacelimbus Group Title

Exactly, and you can verify for yourself that this is equal as writing $\Large \frac{d(y(x+1)e^{-x})}{dx}=xe^{-x}$ It's a bit edgy to multiply it out, but it worked for me.

9. cwrw238 Group Title

right and integrating xe^(-x|) I got -e^(-x)(1 + x) + A

10. Spacelimbus Group Title

same here, so far you have. $\large y(x)(x+1)e^{-x}=-e^{-x}(x+1)+C$

11. cwrw238 Group Title

right so i next divided rhough by e ^(-x) to get y(x+1) = -1((1+x) + Ce^x y(x + 1) = Ce^x -x - 1 and thats my result but the book gives y(x+1) = Ce^x -x + 1

12. Spacelimbus Group Title

hmm I wonder why they divide like that, usually you want to solve a differential equation for the explicit form, therefore solving for y(x), this gives me: $\Large y(x)=\frac{C}{(x+1)e^{-x}}-1= \frac{Ce^{x}}{x+1}-1$

13. cwrw238 Group Title

yea i dont know why they did that

14. Spacelimbus Group Title

makes no sense to me, I just asked WolframAlpha and our answer seems to be correct.

15. Spacelimbus Group Title

what you did seems correct, just seems like a small failure of plus and minus, in the book apparently.

16. cwrw238 Group Title

right thanks

17. cwrw238 Group Title

a typo thanks for your help

18. Spacelimbus Group Title

no problem.

19. cwrw238 Group Title

oh one thing - when you are working out the Integrating factor you dont add a constant of integration right?

20. Spacelimbus Group Title

yes exactly, you can ignore this one during the process (-: