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

Name Enrolment. Roll no. Jinali Makwana 130280103024 303122 Naim Mansoori 130280103025 303123 Madhuri Modi 130280103027 303124 Nayak Shivani 130280103029 303125

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

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

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).

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.

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.

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

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)

i) Solutions : : linearly independent The general solution:

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

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

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

iii) Let The general sol.:

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

Special Case Of Second Order We begin by considering the special case of the second order equation ay’’ + by’ + cy = 0 .. ….(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

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 = 0 …….(2) This equation is called the auxiliary equation of the differential equation (1).

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).

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

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

The general solution will be

Few example based on it :-

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

Few example based on this:

 

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

 

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