Paavai Engineering College Department of Mathematics
UNIT I
ORDINARY DIFFERENTIAL
EQUATIONS
UNIT I 1. 1
,Paavai Engineering College Department of Mathematics
CONTENT
1.1 Higher Order Differential equations with constant coefficients
1.2 Method of variation of parameters
1.3 Method of Undetermined Coefficients:
1.4 Problems based on Cauchy’s Type
1.5 Problems Based on Legendre’s Linear Differential Equation (Equation
Reducible to Linear Form)
.
UNIT I 1. 2
,Paavai Engineering College Department of Mathematics
TECHNICAL TERMS:
1. Differential Equation:
It is an equation that relates one or more unknown functions and their derivatives.
2. Ordinary Equation:
A differential equation containing one or more functions of one independent variable
and the derivatives of those functions.
3. Complementary Function:
It is the general solution of a homogenous linear differential equation.
4. Particular Integral:
The particular integral is the particular solution of the equation:
5. Legendre’s Differential Equation:
A second order ODE has two linearly independent solutions.
6. Cauchy’s Equation:
A linear homogenous ODE with variable coefficients.
7. Variation of Parameters:
It is also known as variation of constants, is a general method to solve in homogeneous
linear ODE.
UNIT I 1. 3
, Paavai Engineering College Department of Mathematics
DEFINITION:
Differential equations which involve only one independent variable are called Ordinary
Differential Equations.
1.1 HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS WITH
CONSTANT COEFFICIENTS.
1.1(a) General form of a linear differential equation of the nth order with constant coefficients is
……..(1)
Where are constants. Such equations are most important in the study of electro –
mechanical vibrations and other engineering problems.
1.1(b). General form of the linear differential equation of second order is
Or Where
Where P and Q are constants and R is a function of x or Constants.
Complete Solution = Complementary Function + Particular Integral
To Find the Complementary Functions:
S.No. Roots of Auxiliary Equation Complementary Functions
1 Roots are Real and Different
2 Roots are real and equal Or
(say)
3 Roots are imaginary
To Find the Particular Integral:
S.No X P.I
1. P.I= ,
2. P.I= , Expand and then operate.
3. Sinax or Cosax P.I=
4. P.I= Replace
UNIT I 1. 4
UNIT I
ORDINARY DIFFERENTIAL
EQUATIONS
UNIT I 1. 1
,Paavai Engineering College Department of Mathematics
CONTENT
1.1 Higher Order Differential equations with constant coefficients
1.2 Method of variation of parameters
1.3 Method of Undetermined Coefficients:
1.4 Problems based on Cauchy’s Type
1.5 Problems Based on Legendre’s Linear Differential Equation (Equation
Reducible to Linear Form)
.
UNIT I 1. 2
,Paavai Engineering College Department of Mathematics
TECHNICAL TERMS:
1. Differential Equation:
It is an equation that relates one or more unknown functions and their derivatives.
2. Ordinary Equation:
A differential equation containing one or more functions of one independent variable
and the derivatives of those functions.
3. Complementary Function:
It is the general solution of a homogenous linear differential equation.
4. Particular Integral:
The particular integral is the particular solution of the equation:
5. Legendre’s Differential Equation:
A second order ODE has two linearly independent solutions.
6. Cauchy’s Equation:
A linear homogenous ODE with variable coefficients.
7. Variation of Parameters:
It is also known as variation of constants, is a general method to solve in homogeneous
linear ODE.
UNIT I 1. 3
, Paavai Engineering College Department of Mathematics
DEFINITION:
Differential equations which involve only one independent variable are called Ordinary
Differential Equations.
1.1 HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS WITH
CONSTANT COEFFICIENTS.
1.1(a) General form of a linear differential equation of the nth order with constant coefficients is
……..(1)
Where are constants. Such equations are most important in the study of electro –
mechanical vibrations and other engineering problems.
1.1(b). General form of the linear differential equation of second order is
Or Where
Where P and Q are constants and R is a function of x or Constants.
Complete Solution = Complementary Function + Particular Integral
To Find the Complementary Functions:
S.No. Roots of Auxiliary Equation Complementary Functions
1 Roots are Real and Different
2 Roots are real and equal Or
(say)
3 Roots are imaginary
To Find the Particular Integral:
S.No X P.I
1. P.I= ,
2. P.I= , Expand and then operate.
3. Sinax or Cosax P.I=
4. P.I= Replace
UNIT I 1. 4
