1. GENERAL SOLUTION. OF HOMOGENOUS LINEAR DIFFERENTIAL EQUATIONS
GENERAL SOLUTION OF HOMOGENOUS LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS

2. Name Enrolment. Roll no. Jinali Makwana 130280103024 303122
Naim Mansoori
303123
Madhuri Modi
303124
Nayak Shivani
303125

3. Included Topics Differential Equation Ordinary Differential Equation
Linear Differential Equation
Auxiliary Equation
General Solution

4. Ordinary Differential Equation:
The differential equation that only involve one independent variable is called ordinary differential equation(O.D.E).
For example:
Only one dependent variable : x

5. Differential Equation:
An Equation containing the derivatives of one or more variables , with respect to one or more independent variables is called a “ differential equation”(D.E).

6. Solution Of Differential Equation
A solution of a differential equation is a relation between the variables which does not involve any derivatives and satisfies the given differential equation.

7. Linear Differential Equation :
Differential Equation is said to be linear if no products exists between the dependent variables and its derivatives, or between the derivatives themselves. In other words all coefficients are function of independent variables.

8. Auxiliary equations
An equation which is obtained by equating to zero the symbolic co- efficient of y is called Auxiliary Equation (A.E.)

9. Constant co efficient Homogeneous:
A, B : numbers (1)
The derivative of is a constant (i.e., ) multiple of
Constant multiples of derivatives of y , which has form
, must sum to 0 for (1)
Let
Substituting into (1),
(characteristic equation)

10. i)
Solutions :
: linearly independent
The general solution:

11. 。 Example 1: Let , Then Substituting into (A), The characteristic equation: The general solution:

12. ii) By the reduction of order method, Let Substituting into (2.4)

13. Choose : linearly independent The general sol
Choose : linearly independent The general sol.: 。 Example 2: Characteristic eq. : The repeated root: The general solution:

14. iii) Let The general sol.:

15. 。 Example 2: Characteristic equation: Roots: The general solution:

16. Special Case Of Second Order
We begin by considering the special case of the second order equation
ay’’ + by’ + cy = ….(1)
where a, b, and c are constants.
If we try to find a solution of the form y = emx, then
after substitution of y = m emx and y = m2 emx , equation (1) becomes
a(m2 emx ) + b(m emx ) + c (emx) = 0 OR
emx ( am2 + bm + c) = 0

17. e mx can’t equal to 0 for all x, it is apparent that the only way y = emx can satisfy the differential equation (1) is when m is chosen as a root of the quadratic equation:
am2+ bm + c = …….(2)
This equation is called the auxiliary equation of the differential equation (1).

18. Since the two roots of (2) are
and there will be three forms of the general solution of (1) corresponding to the three cases:
• m1 and m2 real and distinct (b2 - 4ac > 0),
• m1 and m2 real and equal (b2 - 4ac < 0), and
• m1 and m2 conjugate complex numbers
(b2 - 4ac = 0).

19. Real and Distinct Roots
Case 1:
If all roots are real and distinct then the General solution(G.S) is given by;

20. SOLVED EXAMPLE:
SOLVE :- y’’’ – 6y’’+11y’’-6y =0
Step 1:
the auxiliary equation will be
Step 2:
the auxiliary equation can be factored into
Hence the roots are D=1,D=2,D=3

21. The general solution will be

22. Few example based on it :-

23. Repeated Roots
Case2:
If any two or more roots of an auxiliary equation are equal then the general solution(G.S) is given by:

24. Few example based on this:

26. Complex Conjugated Roots
Case 3:
If roots are in complex conjugate form for example ,then general solution (G.S)
is given by:

29. Thank You