y(0 ' diany ty-o レ 4.(15 pts) Suppose gi(t), va(t) are solutions of (1 + t)y" + (sint)y ey 0. Suppose also that n attains a maximum and y/2 attains a minimum, both at the point t 5. Do y1, /2 form a fundamental set? Explain. y(0 ' diany ty-o レ 4.(15 pts) Suppose gi(t), va(t) are solutions of (1 + t)y" + (sint)y ey 0. Suppose also that n attains a maximum and y/2 attains a minimum, both at...
The Bessel equation of order one-half is X .2 dy d.2 + X dy dar +(x2 - :) y = 0, X > 0 4 a) Verify that yı(x) = x-1/2 sin x is a solution to the equation b) Use reduction of order to find a second linearly independent solution. (Hint: one possibility is y2(x) = x-1/2 cos x.] c) Compute the Wronskian of these two solutions explicitly and verify that it is equal to the solution we computed...
Use Reduction of Order method to find the second linearly independent solution: t2y``- ty`+y = 0. y1=t
If y is a known nonvanishing solution of y" p(t)y + q(t)y 0, then a second solution y2 satisfies 2 У1? where W(y1, y2) is the Wronskian of y1 and y2. To determine y2, use Abel's formula, W(y1, Y2)(t) =C.eJP(t) dt, where C is certain constant that depends on y1 and y2, but not on t. Use the method above o find a second independent solution of the given equation. (х — 1)у" - ху" + у %3D 0, x>...
Two linearly independent solutions of the differential equation y" - 5y' + 6y = 0 are Select the correct answer. a. Y1 = 62, y2 = 232 b. Y1 = 0 -6x, y2 = e** c. Y1 = e-Gx, y2 = et d. Y1 = 0-2, y2 = 2-3x e. Yi = e6x, y2 = e-*
1. Consider the polynomial p.(t)=1+t2 and pz(t)=1 – 12. Is {P1, P2} a linearly independent set in P = Span{1,1, 12? Why or Why not?
2. Let the random variables Y1 and Y, have joint density Ayſy22 - y2) 0<yi <1, 0 < y2 < 2 f(y1, y2) = { otherwise Stom.vn) = { isiml2 –») 05451,05 ms one a independent, amits your respon a) Are Y1 and Y2 independent? Justify your response. b) Find P(Y1Y2 < 0.5). on the
Given the path C: x(t) = (cost, sint, t), 0<t<2n. Let f(t, y, z) = x2 + y2 + 22. Evaluate (12 pts) f(,y,z)ds.
Three linearly independent solutions of the differential equation y'"' - y" - 6y' = 0 are Select the correct answer. a. V1 =e-6s, y2 =xe-1, V3 =1 b. Y1 = 224, y2 = 2-3x, y3 = 1 c. Y1 = 2-6x, y2 = e", y3 = 1 d. Y1 = e3x, y2 = 2-2*, y3 = 1 e. Vi=e , y2=xe-1, V3=1
Two linearly independent solutions of the differential equation y" + 4y' + 5y = 0 are Select the correct answer. a. Y1 = e-cos(2x), y2 = eʼsin (2x) b. Y1 = e-*, y2 = e-S* c. Yi= e-*cos(2x), y1=e-* sin(2x) d. Y1 = e-2xcosx, x, y2 = e–2*sinx e. Y1 = e', y2 = 5x