## lilMissMindset 3 years ago DIFFERENTIAL EQUATION: help please, anybody. (D^2 +4)y =4sin^2 (x)

1. shubhamsrg

y= a cos2x ?

2. shubhamsrg

guess that substn should work..

3. lilMissMindset

aaa. i think i got it. wait, i'll try.

4. shubhamsrg

ohh wait,,i didnt see a "y" also in there. !!

5. Algebraic!

@shubhamsrg I think it's just written in 'operator' notation... $\frac{ d^2y }{dt^2 } +4y = 4\sin^2x$ correct @lilMissMindset ?

6. Algebraic!

heh.

7. shubhamsrg

there's a y also with D^2 .. i had got that,,i didnt see a y being multiplied to whole that time..sorry..

8. Algebraic!

$\frac{ d^2y }{ dx^2 } +4y =4\sin^2x$ rather.

9. lilMissMindset

yes, tat's it

10. lilMissMindset

that*

11. Algebraic!

yes he's correct I believe y =a*cos^2 x

12. Algebraic!

did you get it to work?

13. Algebraic!

that's the particular soln. anyway...

14. lilMissMindset

isn't it like, Acosx+Bsinx+Cxcosx+Dxsinx

15. Algebraic!

why?

16. lilMissMindset

since sinx is twice? applying case 4 of undetermined coefficients

17. lilMissMindset

where it has some repeated imaginary root

18. Algebraic!

look d^2 (a*cos^2(x)) /dx^2 +4a cos^2(x) = 4sin^2(x) second derivative of a*cos^2(x) is 2a*cos^2(x) - 2a*sin^2(x)

19. Algebraic!

this is for the particular solution...

20. Algebraic!

finding the homogeneous solution shouldn't be a problem...

21. Algebraic!

$\lambda ^2 +4 =0$

22. Algebraic!

lambda = 2i

23. Algebraic!

ok?

24. lilMissMindset

i'll work on it. please guide me still, in case i get stuck.

25. Algebraic!

hold on... can you tell me what part is giving you problems?

26. lilMissMindset

getting Yp gives me a lot of trouble. since you told me that it's acos^2 x, i am differentiating it now.

27. Algebraic!

ok:)

28. lilMissMindset

how about if the given is cos^x, then Yp=asin^2 x?

29. Algebraic!

if the forcing term is cos^2 x you mean?

30. lilMissMindset

yes.

31. lilMissMindset

i forgot 2, lol.

32. Algebraic!

yeah, for the same equation, I believe that a*sin^2 x would be the trial solution then...

33. Algebraic!

yes. it would. just checked it.

34. lilMissMindset

thanks!

35. Algebraic!

sure:)

36. lilMissMindset

darn, i can't do it. show me a brief solution please. so i can learn how.

37. lilMissMindset

just for the Yp part. swear.

38. Algebraic!

well, I did it already (a*cos^2(x))" +4a cos^2(x) = 4sin^2(x)

39. Algebraic!

2a*cos^2(x) - 2a*sin^2(x) +4a cos^2(x) = 4sin^2(x)

40. Algebraic!

hmm, I see your point ...

41. Algebraic!

I guess you'll be needing something like yp = a*cos^2(x) + b*sin^2(x)

42. Algebraic!

let me try it out and see...

43. lilMissMindset

sure. :)

44. Algebraic!

seems to work... I got a= 2 b=-2

45. Algebraic!

so 2cos^2 x -2sin^2 x

46. lilMissMindset

answer here in the book is 1/2(1-xsin2x) for the Yp part.

47. Algebraic!

I'll assume that's (1/2)*( 1 - x*sin^2 x )

48. lilMissMindset

yes.

49. Algebraic!

you should check the original, I think you might have written the problem wrong...

50. Algebraic!

you left out and x I think...

51. lilMissMindset

i don't think so.

52. Algebraic!

(1/2)*( sin^2 x - x*sin 2x ) ??

53. Algebraic!

I don't see how (1/2)(1-xsin2x) can work... maybe I'm missing something.

54. Algebraic!

something's off in your original equation or that soln.