Determine which of the following pairs of functions are linearly independent.
1. \(f(x, y)=2 x-4 y-12, \quad g(x, y)=-3 x+6 y+18\)
2. \(f(t)=17 t^{3} \quad, \quad g(t)=e^{x}\)
3. \(f(\theta)=\cos (3 \theta) \quad, \quad g(\theta)=4 \cos ^{3}(\theta)-8 \cos (\theta)\)
4. \(f(x)=x^{3} \quad, \quad g(x)=|x|^{3}\)
5. \(f(t)=e^{\lambda t} \cos (\mu t) \quad, \quad g(t)=e^{\lambda t} \sin (\mu t) \quad, \mu \neq 0\)
6. \(f(t)=4 t^{2}+28 t \quad, \quad g(t)=4 t^{2}-28 t\)
7. \(f(\theta)=\cos (3 \theta) \quad, \quad g(\theta)=4 \cos ^{3}(\theta)-4 \cos (\theta)\)
8. \(f(x)=e^{4 x} \quad, \quad g(x)=e^{4(x-3)}\)
9. \(f(x)=x^{2} \quad, \quad g(x)=4|x|^{2}\)
Determine which of the following pairs of functions are linearly independent.
(1 point) Determine which of the following pairs of functions are linearly independent. NO_ANSWER 1. f(t) = 5t? + 35t, g(t) = 5t2 – 35t NO_ANSWER 2. f(t) = edt cos(ut), g(t) = edt sin(ut) ,70 NO_ANSWER 3. f(x) = 51, g(x) = 5(2-3) NO_ANSWER 4. f(t) = 3t , g(t) = 1t|
Problem 2 Determine if the following functions are linearly independent or linearly dependent. If you believe that they are linearly dependent (i.e. W(5,9) (+) = 0, for all t in some interval) find a dependence relation. 1. f(t) = cost, g(t) = sint 2. f(t) = 61, g(t) = 64+2 3. f(t) = 9 cos 2t, g(t) = 2 cos? t - 2 sinat 4. f(t) = 2t>, g(t) = 14
Question 5 Is the set of functions linearly dependent or linearly independent? f(x) = 7, g(x) = 5x +1, h(x) = 3x2 - 4x + 5 Linearly dependent Linearly independent Have no clue... Question 6 Given a solution to the DE below, find a second solution by using reduction of order. r’y' – 3xy + 5y = 0; y1 = r* cos(In x) y2 = xsin(In x) y2 = x2 sin Y2 = 2 * sin(In) . . y2 =...
(16 points) Determine whether the given set of functions is linearly dependent or linearly independent on the indicated interval. Justify your answers. (a) (8 points) fi(x) = x + 2cos²x, f(x) = 3sin’x, f(x) = x + 2 on (-0,co). (b) (8 points) fi(x) = e34 and 12(x) = e 4x are solutions of the linear homogeneous differential equation y" + y' - 12y = 0 on (-0,co).
Find linearly independent functions that are annihilated by the given differential operator. (Give as many functions as possible. Use x as the independent variable. Enter your answers as a comma-separated list.) 1. D4 2. D2 − 7D − 44 Solve the given initial-value problem: 2. y'' + y = 10 cos 2x − 4 sin x, y(π/2) = −1, y'(π/2) = 0 : y(x)=____________
a) they are linearly independent b)they are linearly dependent c)neither linearly dependent nor linearly independent d)functions cannot be determined in real space x e) none of them (10,00 Puanlar) 2 14,(x) = [1 - Cos(2x)]. uz(x) = Sin?(x) fonksiyonlarının lincer bağımlı yada lineer bağımsız olup olmadıklarını inceleyiniz? a uneer olarak bagimsizdirlar by Lineer olarak bagimlidirlar. Ne lineer bagimline de lineer bagimsizdirlar d Fonksiyonlar, x-reel uzayında belirlenemezdirler c) Hiçbiri Once 2/ Soncalo > Kaput Swim
9. (12 pts) Determine whether the following functions are linearly independent. [ 3e-t [0] 2e (a) xi(t) = -2 , x2(t) = |e3t, x3(t) = 5e L-e- [O] Lel (b) f(x) = 2x2020, g(x) = 2,2020 cos2, h(x) = 2.2020 sinº r.
The indicated functions are known linearly independent solutions of the associated homogeneous differential equation on (0, 0). Find the general solution of the given nonhomogeneous equation. *?y" + xy' + (x2 - 1)y = x3/2; Y1 = x-1/2 cos(x), Y2 = x-1/2 sin(x) y(x) =
(1 point) Are the functions f, g, and h given below linearly independent? f(x) = 621 + cos(9x), g(x) = 621 – cos(9x), h(x) = cos(9x). If they are independent, enter all zeroes. If they are not linearly independent, find a nontrivial solution to the equation below. Be sure you can justify your answer. (e24 + cos(9x)) + (e21 – cos(9x)) + (cos(9.x)) = 0.
(1 point) Are the functions f, g, and h given below linearly independent? f(x) = €3x – cos(4x), g(x) = 23x + cos(4x), h(x) = cos(4x). If they are independent, enter all zeroes. If they are not linearly independent, find a nontrivial solution to the equation below. Be sure you can justify your answer. (e3x – cos(4x)) + (83x + cos(4x)) + (cos(4x)) = 0.