1. IFET COLLEGE OF ENGINEERING
Gangarampalayam, Villupuram – 605 108.
(Accredited by NBA, New Delhi)
Prepared by
T.Soupramanien M.Sc., M.Phil.,
Senior Lecturer / Mathematics
2. Introduction
Definitions
Complementary Solution
Particular Integrals
General Solution
Higher order linear differential equations with constant
coefficients
Method of variation of parameters
Cauchy’s and Legendre’s linear equations
Simultaneous first order linear equations with constant
coefficients Definitions
2
IFET/MATHS/I Year/II Semester/MA2161/Mathematics-II/Ver-1.0
12/10/13
3. Solution of second order linear differential equation with constant
coefficients:
Linear differential equation of nth order with constant
coefficients is defined as:
dny
d n −1 y
d n −2 y
a0
+ a1
+ a2
+ .... + a n y = X
n
n −1
n −2
dx
dx
dx
Where a0 , a1, a2,…,an are constants and X is a function of x.
General solution = complementary function + particular integral
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IFET/MATHS/I Year/II Semester/MA2161/Mathematics-II/Ver1.0
12/10/13
4. Case (i) If m1, m2, m3, …,mn are real & distinct then
C.F =
c1 e
m1 x
+c 2 e
m2 x
+c 3 e
m3 x
+ ... +c n e
mn x
Case (ii) If two roots are equal i.e., m1=m2=m
C.F =
(c1 x +c 2 )e
mx
+c 3 e
m3 x
+ ... +c n e
mn x
Case (iii) If three roots are equal i.e., m1=m2=m3=m
C.F =
4
(c1 x +c 2 x + c3 )e
2
mx
+ c4 e
m4 x
+ ... +c n e
IFET/MATHS/I Year/II Semester/MA2161/Mathematics-II/Ver1.0
mn x
12/10/13
5. Case(iv) If two roots are complex i.e., m1=α+iβ m2=α-iβ
C.F =
αx
(c1 cos β x + c 2 sin β x) e + c 3 e
m3 x
+ ... + c n e
mn x
Case(v) If two pairs of complex roots are equal
i.e.,
m1=m3=α+iβ m2=m4=α-iβ
C.F =
5
αx
((c1 x + c2 ) cos β x + (c3 x + c4 ) sin β x)e + c 5 e
IFET/MATHS/I Year/II Semester/MA2161/Mathematics-II/Ver1.0
m5 x
+ ... + c n e
12/10/13
mn x
