## anonymous 5 years ago How do I solve the differential equation x^2+(x^3+8)y'=0

1. anonymous

Can I divide each term by 1/dy to separate?

2. anonymous

This is in the form m+ny'=0

3. anonymous

see if its exact

4. anonymous

do the partial with respect to y for the m and then do the partial with respect to x in the n portion

5. anonymous

Start by isolating y' .. then integrate both sides

6. anonymous

if they are equal then it is exact

7. anonymous

But to isolate y' I would need to divide each term by y' correct?

8. anonymous

wait...is this cal 1 or differential equations and linear algebra

9. anonymous

Calc II differential equations

10. anonymous

http://1337.is/~gaulzi/tex2png/view.php?png=201103312034348813.png would look like this isolated

11. anonymous

Well you can try implicit differentiation.

12. anonymous

take the derivative of the first term with respect to x

13. anonymous

kristin wouldn't it be -x^2?

14. anonymous

sorry yes, thats correct, minor typo :)

15. anonymous

Ok no problem. Thanks

16. anonymous

I can toss up for you the answer if you want :)

17. anonymous

y' is dy/dx correct? If i were to rewrite it

18. anonymous

yeah

19. anonymous

Ok so if I were to solve y'+y=10 would I just replace y' with dy/dx and then move it all around?

20. anonymous

I'm trying to figure out the steps

21. anonymous

it can be done by the separable variables method

22. anonymous

after some modification you can get dy=-x^2/(x^3+8) dx just integrate both sides

23. anonymous

http://1337.is/~gaulzi/tex2png/view.php?png=201103312042017465.png these are the two answers you can get.. depends if you like to use ln or log

24. anonymous

I know the answer I'm trying to figure out the steps, could you show me the steps kristin? Plz.

25. anonymous

I can show it to you

26. anonymous

did you integrate like she said?

27. anonymous

I know I need to integrate each piece, I just need to know how to get each piece by itself

28. anonymous

do a u substitution

29. anonymous

u=x^3+8 du=3xdu

30. anonymous

du=3x^2 sorry

31. anonymous

you should get: integral( -1/3(1/u) dx

32. anonymous

just take u=x^3+8 --> du=3x^2dx substitute in the integral you will get $y=-1/3\int\limits_{}^{}(1/u)du=-1/3\ln \left| u \right|$

33. anonymous

now just substitute for u=x^3+8 $y=-1/3\ln \left| x^3+8 \right|+c$

34. anonymous

Oh right I got that. Thanks

35. anonymous

np