1. (a) For what values of k does the function= cos kt satisfy the differential equation

49 y''= −16 y?

(Enter your answers as a comma-separated list.)k= _______

(b) For those values of k, verify that every member of the family of functions

y=A sin kt+B cos kt is also a solution.

y=A sin kt+B cos kt y'=A k cos kt−B k sin kt y''= −A k 2 sin kt−B k 2 cos kt.

The given differential equation 49 y''= −16 y is equivalent to

49 y''+16 y=.

Thus, LHS = 49 y” + 16 y = ________

49(−A k 2 sin kt−B k 2 cos kt)+16 (____________)= −49 A k 2 sin kt−49 B k 2 cos kt+(___________) sin kt+16 B cos kt= (16−49 k 2)A sin kt+(___________) cos kt=____________sin c e k 2= ____________.

2.Which of the following functions are solutions of the differential equation''+y= sin x?(Select all that apply.)= cos x= -1/2 x cos xx cos x y=x sin x− 8 x cos x= 1/2 x sin x y= sin x

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