Question

A third-order homogeneous linear equation and three linearly independent solutions are given below. Find a particular solution satisfying the given initial conditions. y (3) + 2y" - y' - 2y = 0; y(0) = 7, y' (0) = 16, y''O) = 0; e y2 = e-X, y3 = e - 2x Y The particular solution is y(x) = .

Answer 1

Que : A third arder homogeneous linear equation and three independent solution are given below. Find a beadsculas sectisfying the given initial conditions linearly solution yBl+ay"-y-ay=0 year=7, y los 10, y como Yi=e", Lazex, Iz = e 2x Solution linearly We have y, ex, Yaz ex and Y2 = 22x are three independent solution of given differential equation Then general solution of given differential equation in y = al gi + Ca ya + C3 Z3 a) y = ciel + cq e" +C 3 ea - y'= ciel - QC² - 203 eax Y" = ciel + caer +4 Cgear By using initial conditions, we get Y(0)27 = a + Q +(z=7

Y(0) = 16 = ci-C2-263 = 16 Y"col = 0 = Cit (2+463 = 0 We write above eystem of Equations in many from as -1 -2 ||C2 - L 1 4 ] [cz] Lol R2 + R2 -4, R₂ + R3 - & [ 17ra 1874 . 2) CitCat C3= 24-362 -9 3C3=-7 7 - cis (ii) (iii) (T) =) -262-3(-3) -93) -26 +7=9 -269-9-7 = 2 1923-1

(i)=) (+(-1) + (-1/2) = 7 == 7+1 +7 • 4 = +3+2 – 3 Therefore particular eolution is y = 22 y + (-1) Yz + (-3) 4