I only need number two solved, please show work, I\'m having trouble understadin
ID: 2865247 • Letter: I
Question
I only need number two solved, please show work, I'm having trouble understading Wroskian. I get 0=-30x-30x
Show directly that the given functions are linearly dependent on the real line. That is, find a nontrivial linear combination of the given functions that vanishes identically. f(x) = 2x, g(x) = 3x^2, h(x) = 5x - 8x^2 f(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) = 3 cos^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 xExplanation / Answer
Accroding to given equations we will set up like that , so you need only questio n2 , so here is the solution
2) We find a nontrivial linear combination c1 f + c2g + c3h of these functions identically
equal to 0. Since all 3 functions are polynomials in x, the function is 0 exactly when
the coefficients on all the powers of x are 0. Since
c1 f + c2g + c3h = c1(5) + c2(2 - 3x2) + c3(10 + 15x2)
= (5c1 + 2c2 + 10c3) + (-3c2 + 15c3)x2,
we require that 5c1 + 2c2 + 10c3 = 0 and -3c2 + 15c3 = 0. From the second equation,
c2 = 5c3. Substituting this into the first,
5c1 + 2c2 + 10c3 = 5c1 + 2(5c3) + 10c3 = 5c1 + 20c3 = 0.
Then c1 = -4c3, and there are no more constraints on the ci. Choosing to set c3 = 1,
c1 = -4 and c2 = 5. We check that this nontrivial linear combination of functions is 0:
(-4)(5) + (5)(2 - 3x2) + (1)(10 + 15x2) = -20 + 10 -15x2 + 10 + 15x2 = 0.