Higher Engineering Mathematics

SECOND ORDER DIFFERENTIAL EQUATIONS (NON-HOMOGENEOUS) 489

I

(vi) The general solution,y=u+v,

i.e. y=Ae^2 x+Be−^3 x− 2 x

−^13 +7 sinx+cosx

Problem 10. Solve the differential equation

d^2 y

dx^2

− 2

dy

dx

+ 2 y=3excos 2x, given that when

x=0,y=2 and

dy

dx

=3.

Using the procedure of Section 51.2:

(i)

d^2 y

dx^2

− 2

dy

dx

+ 2 y=3excos 2x in D-operator

form is

(D^2 −2D+2)y=3excos 2x

(ii) The auxiliary equation ism^2 − 2 m+ 2 = 0

Using the quadratic formula,

m=

2 ±

√

[4−4(1)(2)]

2

=

2 ±

√

− 4

2

=

2 ±j 2

2

i.e.m= 1 ±j 1.

(iii) Since the roots are complex, the C.F.,
u=ex(Acosx+Bsinx).

(iv) Since the right hand side of the given dif-

ferential equation is a product of an expo-

nential and a cosine function, let the P.I.,

v=ex(Csin 2x+Dcos 2x) (see Table 51.1(f)

— again, constantsCandDare used sinceA

andBhave already been used for the C.F.).

(v) Substitutingvinto (D^2 −2D+2)v=3excos 2x

gives:

(D^2 −2D+2)[ex(Csin 2x+Dcos 2x)]

=3excos 2x

D(v)=ex(2Ccos 2x− 2 Dsin 2x)

+ex(Csin 2x+Dcos 2x)

(≡ex{(2C+D) cos 2x

+(C− 2 D) sin 2x})

D^2 (v)=ex(− 4 Csin 2x− 4 Dcos 2x)

+ex(2Ccos 2x− 2 Dsin 2x)

+ex(2Ccos 2x− 2 Dsin 2x)

+ex(Csin 2x+Dcos 2x)

≡ex{(− 3 C− 4 D) sin 2x

+(4C− 3 D) cos 2x}

Hence (D^2 −2D+2)v

=ex{(− 3 C− 4 D) sin 2x

+(4C− 3 D) cos 2x}

−2ex{(2C+D) cos 2x

+(C− 2 D) sin 2x}

+2ex(Csin 2x+Dcos 2x)

=3excos 2x

Equating coefficients of exsin 2xgives:

− 3 C− 4 D− 2 C+ 4 D+ 2 C= 0

i.e.− 3 C=0, from which,C=0.

Equating coefficients of excos 2xgives:

4 C− 3 D− 4 C− 2 D+ 2 D= 3

i.e.− 3 D=3, from which,D=−1.

Hence the P.I.,υ=ex(−cos 2x).

(vi) The general solution,y=u+v, i.e.

y=ex(Acosx+Bsinx)−excos 2x

(vii) Whenx=0,y=2 thus

2 =e^0 (Acos 0+Bsin 0)

−e^0 cos 0

i.e. 2 =A−1, from which,A= 3

dy

dx

=ex(−Asinx+Bcosx)

+ex(Acosx+Bsinx)

−[ex(−2 sin 2x)+excos 2x]

When x=0,

dy

dx

= 3

thus 3 =e^0 (−Asin 0+Bcos 0)

+e^0 (Acos 0+Bsin 0)

−e^0 (−2 sin 0)−e^0 cos 0

i.e. 3 =B+A−1, from which,

B=1, sinceA= 3

Hence the particular solution is

y=ex(3 cosx+sinx)−excos 2x