Verify every number of a family of functions is also a solution

  • #1

Homework Statement


(a)For what values of k does the function y=coskt satisfy the differential equation 4y''=-25y?
(b) For those values of k, verify that every number of the family of functions y=Asinkt+Bcoskt is also a solution.

The Attempt at a Solution


(a) y=coskt, y'=-ksinkt, y''=-k2coskt.
4(-k2coskt)=-25coskt; 4k2=25, so k=+/-5/2
(b) y= Asinkt+Bcoskt; y'= +/-Akcoskt+/-BKcoskt; y''= +-Ak2sinkt+/-bk2coskt
Im not sure how to verify this...
 

Answers and Replies

  • #2
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,728

Homework Statement


(a)For what values of k does the function y=coskt satisfy the differential equation 4y''=-25y?
(b) For those values of k, verify that every number of the family of functions y=Asinkt+Bcoskt is also a solution.

The Attempt at a Solution


(a) y=coskt, y'=-ksinkt, y''=-k2coskt.
4(-k2coskt)=-25coskt; 4k2=25, so k=+/-5/2
(b) y= Asinkt+Bcoskt; y'= +/-Akcoskt+/-BKcoskt; y''= +-Ak2sinkt+/-bk2coskt
Im not sure how to verify this...
Well, do you have 4y'' = -25y or not?
 
  • #3
I believe you would if y''=-Ak2sinkt-Bk2coskt?
 
  • #4
HallsofIvy
Science Advisor
Homework Helper
41,833
956
Did you miss where the problem says "for those values of k"? You are asked to show that y= A sin((2/5)t)+ B cos((2/5)t) and y= Asin((-2/5)t)+ Bcos((-2/5)t) both satisfy [itex]4d^2y/dx^2= -25y[/quote] for any A and B.
 
  • #5
ok so (for a positive k) y= Asinkt+Bcoskt; y'= -Akcoskt+BKsinkt; y''= -Ak2sinkt-bk2coskt
so,
-4[A(5/2)2sin(5/2)t-B(5/2)2cos(5/2)t= -25[Asin(5/2)t+Bcos(5/2)t
-25Asin(5/2)t-25Bcos/5/2)t= RHS
 
Last edited:

Related Threads on Verify every number of a family of functions is also a solution

Replies
2
Views
8K
Replies
2
Views
831
Replies
0
Views
979
Replies
2
Views
14K
Replies
2
Views
13K
Replies
0
Views
652
Replies
4
Views
4K
Replies
6
Views
8K
Top