This document discusses linear differential equations of second order with constant coefficients. It presents: 1) The general form of a second order linear differential equation with constant coefficients. 2) Methods for finding the complete solution, which is the combination of the complementary function (C.F.) and particular integral (P.I.). 3) Solutions for three cases of the auxiliary equation: when roots are real and distinct, real and equal, and imaginary. Worked examples are provided to illustrate each case.

1. K.Muthulakshmi M.Sc., M.Phil.,
Assistant Professor
V.V.Vanniaperumal College for Women,
Virudhunagar

2. 1

3. Ordinary Linear Differential
Equations of Second Order
2
• Linear differential equation of Second order
with constant Coefficient
• Linear differential equation of Second order
with variable coefficient

4. INTRODUCTION
⚫ General FormofLinear Differential Equa
tion
⚫ WhereP andQ botharecons
tants
(independent variable)
d
3
2
y d y
d x 2
d x
 P  Q y  R
⚫ R is the function ofx
or zeroor constant

5. • Solution of linear Differential Equation of Second
Order with Constant Coefficient
dx2
dx
d2
y

dy
P Qy  R
Nowwe can write as above D.E
D 2
y  PDy  Qy  R
4
D 2
 PD  Q y  R

6.  Complete (General) Solution of Differential Equation
dx 2
dx
 General Sol.= Complementary Function + Particular Integral
Y = C.F. + P.I.
5
d 2
y

dy
P  Qy  R

7. Method of ComplementaryFunction(C.F
.)
6
D2
 PD  Qy  R.........(1)
Replace D by m in Equation (1)
Auxiliary Equation (A.E.)
m2
 PmQ  0
Case-I when the roots of A.E. are real and distinct
Let
Let
m m1,m2

8. Then
Where C1 and C2 are arbitrary Constants.
Example: Solve the given differential equation
Solution: The given equation is
Auxiliary Equation
7

9. Here Particular Integral = 0 (because R=0)
8
Now General solution Y = C.F. +P.I.
Where C1 and C2 are arbitrary Constants.

10. Example: solve the given differential equation
d 2
y d y
d x 2
d x
 3  5 4 y  0
Solution: The given equation is
A.E.
Or
Now General Solution Y = C.F. + P.I.
9

11. Case-II When the roots of A.E. are real and equal
Let
Then
Where C1 and C2 are arbitrary Constants.
Example: Solve the given differential equation
Solution: The given equation is
10

12. Here Auxiliary Equation is
Or
General Solution Y = C.F. + P.I
Where C1 and C2 are arbitrary Constants
11

13. Case-III when the roots of A.E. are Imaginary (Complex)
Let
Then
If
Then
Where C1 and C2 are arbitrary Constants
12

14. Example: solve the given differential equation
Solution: The given equation is
Auxiliary Equation is
Now General Solution Y = C.F. + P.I.
13

15. Example-: Solve the given differential equation
Solution: The given equation is
Here A.E. is
Now General Solution Y= C.F. + P.I.
Where C1 and C2 are arbitrary constants.
14