complementary function for same roots, distinct roots, imaginary roots.

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
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• 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
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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.
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d 2
y

dy
P  Qy  R

7. Method of ComplementaryFunction(C.F
.)
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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
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9. Here Particular Integral = 0 (because R=0)
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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.
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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
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12. Here Auxiliary Equation is
Or
General Solution Y = C.F. + P.I
Where C1 and C2 are arbitrary Constants
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13. Case-III when the roots of A.E. are Imaginary (Complex)
Let
Then
If
Then
Where C1 and C2 are arbitrary Constants
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14. Example: solve the given differential equation
Solution: The given equation is
Auxiliary Equation is
Now General Solution Y = C.F. + P.I.
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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.
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