Page 119 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation Sachin Kaushal, Lovely Professional University

Notes Unit 7: General Properties of Solutions of Linear
Differential Equations of Order n

CONTENTS

Objectives
Introduction
7.1 Existence and Uniqueness of the Solution of a System of Differential Equations
7.2 General Properties of Solution of Linear Differential Equations of Order n
7.3 Solution of the Linear Equation with Constant Coefficients

7.4 Particular Integral
7.5 Summary
7.6 Keywords
7.7 Review Questions
7.8 Further Readings

Objectives

After studying this unit, you should be able to:
 Deal with a differential equation of order n, and there are lots of properties to be kept in
mind before actually solving any problem.
 Discuss Picard method of existence and uniqueness of the linear differential equation
before solving any problem.
 Know some properties of linear differential equation of nth order with constant coefficients
and the solutions obtained both for complementary functions (C.F.) and Particular Integral
(P.I.)

Introduction

The method of proof of the existence of the solution of nth order differential equation is similar
to that of first order one.
Some properties of the differential equations are listed and later used to find the solutions of a
class of nth order differential equations.
7.1 Existence and Uniqueness of the Solution of a System of

Differential Equations

An nth order linear differential equation involving dependent variable y and independent
variable x can be written as

n
d y d n 1 y d n 2 y
a n a n 1 a n 2 ... ay = 0
dx n dx n 1 dx n 2

112 LOVELY PROFESSIONAL UNIVERSITY