Page 130 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

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

will have the following properties. Notes
Nature of the roots Solution
1. Real and distinct y c e m 1 x c e m 2 x ... c e m n x
2
n
1
i.e., m m 2 ,...m
,
n
1
2. Real and equal, each m1 (say) y (c 1 c x c x 2 ... c x n 1 )e m 1 x
2
n
3
3. Non-repeated roots as i y (c 1 cos x c 2 sin x e
ax
)
or y c e x cos( x c 2 )
1
4. Repeated roots i , r times y [(c 1 c x ... c x r 1 ) cos x (c 1 c x ... c x r 1 )
2
r
2
r
]
sin x e x
h
5. Irrational roots as y c e ax cos (x c 2 )
1
or y c e x sin (x c 2 )
h
1

d d n
n
Example 4: The symbol D is used for for D for n . It should be kept in mind that
dx dx
1
1
D and D are the inverse operations, i.e., as D means differentiations, D means integrations.
Illustrative Examples
2
d y dy
Example 1: Solve: 2 7 44y 0.
dx dx
2
Solution: The equation can be written as (D 7D 44) y = 0
The auxiliary equation is
m 2 7m 44 = 0 or (m 11) (m + 4) = 0
m = 11, 4, which are real and distinct. Hence solution of the given equation is
y = c e 11x c e 4x .
1 2
2
d y dy
Example 2: Solve: 2 4 y 0.
dx dx
Solution: The given equation is
2
(D 4 D + 1) = 0
The auxiliary equation is
2
m 4 m + 1 = 0
4 16 4
m = 2 3
2
Hence general solution is

y = c e (2 3)x c e (2 3)x
2
1
It can also be written in the form
y = e x (c e 3x c e 3x )
1 2

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