(a) For what values of k does the function y = cos kt satisfy the differential e

Question

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

is also a solution.

LHS = 16y'' + 49y = 16(?Ak2 sin kt ? Bk2 cos kt) + 49 = ?16Ak2 sin kt ? 16Bk2 cos kt + sin kt + 49B cos kt = (49 ? 16k2)A sin kt + cos kt =     since k2 = . For what values of k does the function y = cos kt satisfy the differential equation 16y'' = ?49y? 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' = Ak cos kt ? Bk sin kt y'' = ?Ak2 sin kt ? Bk2 cos kt. The given differential equation 16y'' = ?49y is equivalent to 16y'' + 49y = . Thus, LHS = 16y'' + 49y = 16(?Ak2 sin kt ? Bk2 cos kt) + 49 = ?16Ak2 sin kt ? 16Bk2 cos kt + sin kt + 49B cos kt = (49 ? 16k2)A sin kt + cos kt = since k2 = .

Explanation / Answer

a')Differentiating the function twice with respect to t:

y = cos(k*t)

y' = -k*sin(k*t)

y'' = -(k^2)*cos(k*t)

Plug this back into the D.E.:

-16*(k^2)*cos(k*t) = -49*cos(k*t)

16*k^2 = 49

k^2 = 49/16


k = +7/4 and k = -7/4


k = +sqrt(49/16) and l = -sqrt(49/16)

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