Question

(10) 3. Solve the ODE byhenethod of undetermined coefficients 1" - 4y - 12y = re

Answer 1

(3) Solution. 9 5 Then, DE d dhe -) mo 2, D D o The given ode is 42 8" - 47' - 124 102-40-12)y=ress, xe or where o² = d² dx² correspondinga auxiliary equation is m2 4m ~12 O m2 -6m +2m~12 = 0 am (m~6) + 2(m-6)=0 3 m~6) (M+2) = 0 6 Complementary function of given ode is Cie + C, e where are arbitrary constants Let the particular be of the forum (An+ B) een where А coefficients to be determined. (Ax+B) et Now 40 » y(x) = 4(A2+ B) 64 x Ае = A 2+A+48)244 =) Up" (*) 414AX + A+4B) e + 4 A e -20 6a Yo (a) - C1, C2 9 integral ) and are 48 Up (a): + 4x 4x

= ode Now, (16 AX + 8 A +166) e4a Yo satisfier goiven .! Yo" - 448'-124 (16 A & + 8A +16 B)e49 4(444+A+48)644 -12 (A2+B) 44 > xela 4x xe = -) = I 36 4x 1 = (-12A2 +4A - 12B )2 4* xe 40 -12 A = 1,4A -12 B =0 => A = - = ; B = ŽA .! Yo(a) (Aa +Be (-ix-35) 047 (3x + 1) the general jhemeroal solution of goiven ode is = yc (2) + yo(a) ci e 48 e 36 Hence, g 62 + C2 e e (32+1), -22 36 where Ci, C, are arbitrary constants,