Page 128 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 7: General Properties of Solutions of Linear Differential Equations of Order n

This will be the case when all the roots, m , m , m ,... m of the auxiliary equation are real, distinct Notes
1 2 3 n
and different.
Auxiliary Equation having Equal Roots

If the auxiliary equation has two equal roots, say m and m , the solution of the given equation
1 2
n
d y d n 1 y
P P ... P y = 0
0 n 1 n 1 n
dx dx
will be y = (c c )e m 1 x c e m 2 x ... c e m n x
1 2 3 n
or y = ce m 1 x c e m 2 x ... c e m n x
3 n
where c + c = c.
1 2
This is not the general solution of (i), because it contains (n 1) arbitrary constants while the
order of the equation is n. To obtain the general solution of (i) in this case, we proceed as follows:

dy 2
2
Consider the repeated factor as m 1 y 0. This can be written as (D m 1 ) y 0,
dx
d
where D = .
dx
Put (D m ) y = ;
1
then (D m ) = 0.
1
d
Therefore = m 1
dx
d
or = m dx
1

Integrating, we have log = m x
c 1
2

Hence = c e m 1 x .
2
or (D m ) y = c e m 1 x
1 2
dy
or m y = c e m 1 x
1
2
dx
This is a linear differential equation and we will have
ye m 1 x = c 1 c e m 1 x . e m 1 x dx
2
= c 1 c x
2
y = (c 1 c 2 ) x e m 1 x .

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