Question

(i) Use the inverse D-operator method to solve (i) (D2 – 3D + 5)y = €22. (ii) (D2 + 3)y = sin —2x.

Answer 1

Auxiliary equation m² +p, ma P2 =0 XN 12 If root mazm, then on cfaçem tem mia mza If root mam, & then cfa ot If root afiß then c.fi= ex (q. Sin Bx + q Cospe)

(1) Solution: the given ODE is ²- 3D+5y = 2 To find C.F. to the auxiliary exquation is m²-3m45=0 ..m=t3)2 Vert - 4x145 3219-20 2 = 3 + 5-11 - 3 2 1 1 2 2 C.F. = ² (a Cos IT & & & SinnTix) To find P.I PI = D²_3D45 Where D? 3pag is inverse sparator, - e22 Formula if fragto f(09 4-645 if f(a)=0 f(D+a) 9-6

The general solution is y=cof. & P.I. g=es (acast Tix &G Sin iTix) + Where a,e constant. are arbitrary

(17) Solution: The giver ODE is (1+3) y = Sin (-22) To find cof The auxiliary equation is m²+3=0 m²=-3 m=253 m=oIBi C.F. = eix (a Cos &3x + q Sin $3x) To find P.I. 2.2.- des Sin (23) P.I.= D whora Wheon D33 2 + where we is invas 3 - is inverse oparator. Then Here a=-2 careful a P.I. = 72 Sin (2x) - Sin (-2x) -(-2 +3 Formula FCO2) Sinax = Sinax ff-az) if ffra?)=0 = Sin(+22) - 4 +3 - Sin (-20) = - Sin (-22)

The general solution is y = C.F. of P. I > y = ex (e cosbx tQ Sin 13x) - Sin (-28) y = Ci Cos 3x + Sin 3x - Sin2x). Where ca ,C₂ arbitrary constant