Question

(17 points) (a) Find the general solution of the differential equation y" (t) + 4y(t) = 0. general solution = (Use the letters A and B for any constants you have in your solution.) (b) For each of the following initial conditions, find a particular solution. (i) y(0) = 0, y'(0) = 1: y = (ii) y(0) = 1, y'(0) = 0:y= (iii) y(0) = 1, y(1) = 0:y= (iv) y(0) = 0, y(1) = 1: y = (On a sheet of paper, sketch a graph of the solutions you found in part (b) to see how they differ.)

Answer 1

Soln! - y cd) + 48 (A) = 0 D2 + 4 = 0 D = I21 General soln + So Generd soln y (d) = Cixit cixit +G2it = a, (co 8220 tisinca) +C₂ (Coord-isind YA) Cosat (CAG) + sinad (i(e-ca] Domine C Cosat e sint (d'acita C=icance Y(A) = 'cosse + " sinat, when clich (6) For particulą soln Il y (at) z c' cosat + sinza y (o)= c' 0 = c ) d'ao y' (t) = 20c-sinat) + 20" Coozt y'co) = +ad! ac" are content 4 o "=ť 1 + ad! o = 20 (A) = 1 sinnt. y (0) = c' to d'y dzi yl(o) = 0 - Czo y(t) = Cosat y(o)- d to = c = c'-| y' (1) = 20 (-sin2] +2c" cosa 0 =-2 sinn & ach Cosz = d'-esin-tana Y(A) = Coxat & tan sinad 12 Cosa