(1 point) Calculate the Wronskian for the following set of functions: f1(x) = 0, f2(2) =...
Determine whether the given set of functions is linearly independent on the interval (−∞, ∞) f1(x) = x f2(x) = sin(x) f3(x) = sin(2x)
2. (a) Let B = {f1, f2, f3} be a subset of P2 where fi(x) = x² – 3, f2(x) = x2 – 2x and f3(x) = x. Show that B is a basis of P2. (b) Determine whether or not the following sets are subspaces of F. (i) X = {f € F | f(x) = a(x + cos x), a € R}. (ii) Y = {f EF | f(x) = ax + sin x, a € R}. (c)...
step by step please Without the aid of the Wronskian, determine whether the given set of functions is linearly independent or linearly dependent on the indicated interval. (a) (%) = In x, f(x) = Inx",(0, ) linearly dependent linearly independent (b) (x) = 2x",,(*) - +, n = 1, 2, Vdots, (-,) linearly dependent linearly independent (c) (X) = x, f(x) - X+4.(-.,-) linearly dependent linearly independent ,) (d) {(x) = cos(x + 2), f(x) - sinx, ( linearly dependent...
17. Another way to check if y1, y2 are linearly INDEPENDENT in an interval I is: for all I for all I does not exist for all I d. none of the above 18. If y1 is a solution of the equation y "+ P (x) y '+ Q (x) y = 0, a second solution would be y2 (x) = u (x) y1 (x) where u (x) it is: d. all of the above 19. The following set...
3. (10 points) Let F denote the vector space of functions f: R + R over the field R. Consider the functions fi, f2. f3 E F given by f1(x) = 24/3, f2() = 2x In(9), f() = 37*+42 Determine whether {f1, f2, f3} is linearly dependent or linearly independent, and provide a proof of your answer.
(1) Calculate the Wronskian of the following vectors and determine if they are pointwise linearly independent or dependent. e 0 0 y(1) ). y (2) y (3) 3e- 3e24 6e2.c 2e34 0 W(y(1), y(2), y(3) Circle One: Independent Dependent
Are the functions fi (x) = ex+4 and fz(x-er-5 linearly dependent or independent? A. Linearty dependent OB. Linearly independent Which of the following best describes the correct choice for part (a)? (Carefull) 0 A. Since the only solution to cfı + c/2 = 0 is ci = c2-0. B. Since the Wronskian equals zero for at least one x on (-o, o). C. Since the Wronskian never equals zero on (-oo, oo). D. Since the functions are scalar multiples of...
Consider the following functions. fy(x) = x, fz(x) = x2, f3(x) = 2x - 4x2 g(x) = C7f1(x) + c2f2(x) + c3f3(x) Solve for Cy, cy, and cz so that g(x) = 0 on the interval (-00, 00). If a nontrivial solution exists, state it. (If only the trivial solution exists, enter the trivial solution {0, 0, 0}.) {C1,C2,C3}={C } Determine whether f1, f2, fz are linearly independent on the interval (-00, 0). O linearly dependent O linearly independent
(Part b) Write the vector sum F1+F2 +F3 in terms of the unit vectors i and j. Use Fi = Fi), F, = |F2), and F3 = |F3| to be the magnitude of the vectors Fi, F2, and F3, respectively. Drag the appropriate terms to construct the correct expression. Pay attention to the difference between a and in the trigonometric functions. Ē + F2 + Fz (F1 + F2 + F3 + + F2 + F3 · α E +X...
Compute Laplace transforms of the following functions: (a) f1 = (1 + t) (b) f2 = eat sin(bt) 11, 0<t<1, (c) f3 = -1 1<t<2, | 2, t>2, Find the functions from their Laplace transforms: (a) Lyı] s(s + 1) (s +3) 2+s (b) L[42] = 52 + 2 s +5 (c) L[y3] = Solve the following initial value problems using the Laplace transform. Confirm each solution with a Matlab plot showing the function on the interval 0 <t<5. (a)...