Show directly that the given functions j are linearly dependent on the real line
ID: 3120488 • Letter: S
Question
Show directly that the given functions j are linearly dependent on the real line. That is. find a non-trivial linear combination of the given functions that vanishes identically. f(x) = 2x, g(x) = 3x^2, h(x) = 5x - 8x^2 (x) = 5, g(x) = 2 - 3x^2, h(x) = 10 + 15x^2 f(x) = 0, g(x) = sin x, h(x) = e^x f(x) = 17, g(x) = 2 sin^2 x, h(x) = 3cos^2 x f(x) = 17, g(x) = cos^2 x, h(x) = cos 2x f(x) = e^x, g(x) = cosh x, h(x) = sinh x In Problems 7 through 12. use the Wronskian to prove that the given functions are linearly independent on the indicated interval.Explanation / Answer
2) f(x) =5, g(x) = 2-3x2,h(x) = 10+15x2
4f(x) - 5g(x) - h(x) = (4*5) - (10-15x2) - (10+15x2)
=20 - 10 +15x2 -10 - 15x2 =0
3) f(x) = 0, g(x) = sinx, h(x) = ex
hx) = ex, g(x) = sin x are linearly independent
4) f(x) =17, g(x) = 2sin2x,h(x) = 3cos2x
sin2x + cos2x = 1
(17/2) g(x) + (17/3) h(x) = 17sin2x + 17cos2x = 17
f(x) - (17/2) g(x) - (17/3) h(x) = 0
6) f(x) = ex, g(x) = coshx, h(x) = sinhx
cosh(x) = (ex+e-x)/2.
sinh(x) = (ex-e-x)/2
cosh(x)+sinh(x) = ex
g(x) + h(x) -f(x) = 0