Question

If S= {e*, e*, xe* is a fundamental solution set for the homogenous differential equation with constant coefficients dy a2 dy +a dx2 d y + a y 0, dx dx find the values of a ,i= 0, 1, 2, 3 (5 markah/marks) Dengan nilai yang diperolehi dari (a), selesaikanlah masalah nilai awal yang berikut: (b) day 2 (1+2e) d2 y a a dx dy ax a ax dengan 0)1,y' (0)- 2,y (0)-3. By using the values obtained in (a), solve the initial value problem: d2 y dy ay 2 (1+2e ) d'y a d x * ax ax with 0)1,y' (0) 2,y (0) =3.

Answer 1

+a,9=0. D-t a, D+0, D a. Le, e'e Solution i Rots of the equedi0, is -than 40= e It Cast%)- PpM Saluhion dat) b Capariny atTon M=l. m -\, \,\ So, Rasts f te eution dfevestid etioni, (D+) (o-)*=0 ave (D Co dy A b Cparing euotion & a3=, a,

= a (1t e) Gete (de -D+1 DD-Dtl DD-Dt 0D-Ot 2e denonun became S D2-D-Dt! 2x again denotunatov 6 D- Zre. (+ ae = 2+xe* 2 X = e(tC) e e+e 72

yo=+G G+Gm+x¢c* a=eestaJ '=C st2+2] +Fst+srju#j = Gte acstta ++ca) n+n°| Subhtate akre vaseg in y, y, y'tu CrtCLETI yo)= 2tC tG = } >€ CtCt3=2 0)CaC3tCz+2=3 =) Gt33= | Balring egitions, CtCt201= -(+20>/ tC tG= \ }- =b Rna Resott fom equaton T