Question

Consider the following differential equation:

4y(4) + y" - 18y' + 13y = et

a) Knowing that r1 = 1 is a double root, find the other two roots.

b) Find the corresponding complementary solution yc(t).

c) Find the corresponding particular solution yp(t).

Answer 1

4 M + 4" - 18y 13y=et @ auxilary equation of homogenous equation of ① uma + m² 18m +13 = 0 (9) m = 1 is a root find others by long division 4m² +4 m² +5m-13 m-il um"+ m²-18m +13 ЧmЧ . 4m² + 4m² Elt w8i- zuh-suh 5m² -18m +13 5 m² - 5m -13m +13 -13m +13 + (m-1) (4 m² +4 m² +5m-13 m=1 81 double root a So we again use long division

4m² 8m +13 mail Ym3 +4m² + 5m= 13 4 m² - 4m² 2 8 m² +5m-13 8 m²-8m 13m - 13 13m-13 + el - wil Elt. ustun consider 4m² +8m +13 use quadratic formula -6 I Seyac - 8 I J 64-4 (4) (13 a 29 64-208 89 J-144 8 + 127 8 + 3 2 - - II @1.5;

Roots are m=11_-1+11, -2 (b) Yo = (( + Cax) ex + e* C3 Sin (3x) + Cu Cos (3x)) complementary Solution Uc function is So UC set 18 {exy But It thetudes me But et is a solution of the congesponding homogeneous UC set by a differential equation So multiply Now uc set is şxex But again xet is a solution of the corresponding homogenous differential equation So multiply o UC set by x² Now UC set 18 { x² et} Let Yp = A x²e²

x Let yp = A x² e Y'p = A lx x² et 2x e y' p = x² et + 2x.ex + 2xiex + 2oces + ze A ne² e AC x² 2 4x ett 2ex) + yp= c² et + axet + 4 xe" +40² +2e 6ex) 7 ze AL - 6x et + + A oc² et + Alx et + 2xe? sxex + 6x et text ten + 12e) - put in I +48A ex H 4 A x² e? A 32 xe + A ² et + 4 Axel + 2A et -18Aziet - 36 Axe + 13 A2²ex => 5oA ex et 50 A=1 A= 1 50 Up= I x²ex 50