Question

Find the general solution of the second order constant coefficient linear ODEs

7. Find the general solution of the second order constant coefficient linear ODE. (a) y" +2y = 0 (b) 2y" – 3y +y=0 (c) y" – 2y – 2y = 0 (d) y" – 2y + 2y = 0 (e) y" + 2y - 8y = 0 (f) y" +9y=0 (g) y" – 4y + 4y = 0 (h) 25y" – 10y' +y=0

Answer 1

a) y "+2y = 0 It quxillan equation is, m²+2m = 0 => mcm+2)=0 hab m = 0, -2. As roots are real & distinct, General solution is onl no Y = Cieort C2 e 2x 1 5 Y = G + en 24 is, b] 27"-37' + y = 0 The auxillary equation am²-3m + 1 =0 → 2m2 -2m-m +=0 → 2m (m-1)-1(m-1)=0 > (2m - 1) (m-1) = 0 m = 1 1 21 As roots are real & distinct, General solution is y = at Get

c) The y"-27'-240 aurollary equation is, m²-2m- 2 = 0. m = -(-2) + √(-2) 2_401) (-2) m ayo C 2017 Cts Ctmm m = 27 J13 2 m = 24 13 2-1 13 boonoo tood 2 2 notube lonan As roots are real & distinct, General solution ist (2t 13 x 1 Y = G è ? + Co | 0 The + 2 + 2 = 0. auxillan equation is, m²_2m +2=0 m = -(-2) + √(-2)² 4 (1) (2) 2010 m= 2 & J-4 - 2 + 21 c Iti > As roots y = are complex, general solution is e? [c, cosx + Casinx]

e The yllt 24' - 84 = 0 auxillary equation is, Door m²+2m - 8 = 0 m² +4 m-am - 8 = 0 mcmt4) -2 (m+4)=ons (m+4) (m-2) ao m = -4,2 . As roots are read & dist » General solution is, 7. Сре41 +С, е? | f ] YM+gy =o The auxillary equation is, m²+9=0 > m=13 L As roots are complex, General solution is, Y c = cos3x + Casinax

g y"-40'+ 4y =o. The ausullay equation A, m²-4m +4=0 ² (m. 2)² = 0 => m= 2,2 As roots are real & repeated, solution is, y = (a + Cax) eza hl 254"-10y' +y = 0 The auxillan equation is, 25 m2 -lom +120 + m2- = 0 25 92 5 25 m = 1 . As roots solution is, are real & a repeated, ya (Ct Cad) et