Higher Engineering Mathematics

482 DIFFERENTIAL EQUATIONS

Table 51.1 Form of particular integral for different functions

Type Straightforward cases ‘Snag’ cases See

Try as particular integral: Try as particular integral: problem

(a)f(x)=a constant v=k v=kx(used when C.F. 1, 2

contains a constant)

(b)f(x)=polynomial (i.e. v=a+bx+cx^2 + ··· 3

f(x)=L+Mx+Nx^2 + ···

where any of the coefficients

may be zero)

(c)f(x)=an exponential function v=keax (i)v=kxeax(used when eax 4, 5

(i.e.f(x)=Aeax) appears in the C.F.)

(ii)v=kx^2 eax(used when eax 6

andxeaxboth appear in

the C.F.)

(d)f(x)=a sine or cosine function v=Asinpx+Bcospx v=x(Asinpx+Bcospx) 7, 8

(i.e.f(x)=asinpx+bcospx, (used when sinpxand/or

whereaorbmay be zero) cospxappears in the C.F.)

(e)f(x)=a sum e.g. 9

(i)f(x)= 4 x^2 −3 sin 2x (i)v=ax^2 +bx+c

+dsin 2x+e cos 2x

(ii)f(x)= 2 −x+e^3 x (ii)v=ax+b+ce^3 x

(f)f(x)=a product e.g. v=ex(Asin 2x+Bcos 2x) 10

f(x)=2excos 2x

51.3 Worked problems on differential

equations of the form

a

d^2 y

dx^2

+b

dy

dx

+cy=f(x)where

f(x)is a constant or polynomial

Problem 1. Solve the differential equation

d^2 y

dx^2

+

dy

dx

− 2 y=4.

Using the procedure of Section 51.2:

(i)

d^2 y

dx^2

+

dy

dx

− 2 y=4 in D-operator form is

(D^2 +D−2)y=4.

(ii) Substitutingmfor D gives the auxiliary equa-

tionm^2 +m− 2 =0. Factorising gives: (m−1)

(m+2)=0, from whichm=1orm=−2.

(iii) Since the roots are real and different, the C.F.,
u=Aex+Be−^2 x.

(iv) Since the term on the right hand side of the given

equation is a constant, i.e.f(x)=4, let the P.I.

also be a constant, sayv=k(see Table 51.1(a)).

(v) Substituting v=k into (D^2 +D−2)v= 4

gives (D^2 +D−2)k=4. Since D(k)=0 and

D^2 (k)=0 then− 2 k=4, from which,k=−2.

Hence the P.I.,v=− 2.

(vi) The general solution is given byy=u+v, i.e.

y=Aex+Be−^2 x− 2.

Problem 2. Determine the particular solu-

tion of the equation

d^2 y

dx^2

− 3

dy

dx

=9, given the

boundary conditions that whenx=0,y=0 and

dy

dx

=0.

Using the procedure of Section 51.2:

(i)

d^2 y

dx^2

− 3

dy

dx

=9 in D-operator form is

(D^2 −3D)y=9.