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Chapter 1 Introduction to Differential Equations1.1 Introduction The mathematical formulation problems in engineering and science usually leads to equations involving derivatives of one or more unknown functions. Such as equations are called differential equations.

X(t)

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B. Ordinary differential equation (ODE’s): The equation contains ordinary derivatives with respect to a single independent variable.

C. Partial differential equation (PDE’s): The equation contains partial derivatives with respect to two or more independent variables.

2 2 2

2 2 2 0u u ux y z

1.2 Definitions A. Differential equation: An equations containing one or more derivatives of the function under consideration.

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2 ( )d xm kx F tt

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E. Solutions: Any function f, defined on an interval I and possessing at least n derivatives that are continuous on I, which when substituted into an nth-order ordinary differential equation reduces the equation to an identity, is said to be a solution of the equation on the interval. Types of solution include:1. General solution, 2. Particular solution, 3. Singular solution.

D. Order: The highest order of derivative in the differential equation

' '' ( )( , ( ), ( ), ( ),..., ( )) 0nF x u x u x u x u x

D’. Degree: The highest power of the highest order of derivative in the differential equation.

2 2 22

2 2 ( ) 0z z zx y x y

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1. General solution: A n-order differential equation F(x, y, y’,…,y(n))=0 The solution can be expressed as y=g(x,c), c is a constant. The solution is called the general solution of the ODE.

2

2 0 2

dy x dy xdxdxy x c

2. Particular solution: For a particular c1, the solution can be expressed as y = g (x, c1) which is called the particular solution.

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3. Singular solution: In some cases there may be further solutions of a given equation which cannot be obtained by assigning a definite values to the arbitrary constant in the general solution; such a solution is called a singular solution of the equation.

2 0y xy y The equation has the general solution

representing a family of straight lines, where each line corresponds to a definite value of c. A further solution is

which is a singular solution.

2y cx c

2

4xy

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Boundary-value problem: Conditions specified at both ends are called boundary conditions, and the differential equation together with the boundary conditions is called aboundary-value problem.

Considering a function T( , ); at =0, T(0,t) = a and T(L, t)=sint

x t x

F. Initial-value problem: Conditions specified at a single point, are called initial conditions, and the differential equation together with those initial conditions is called an initial-value problem.

0

0

For a function ( , ); at t=0, y( , 0) = y ( ), 0, (0)

y x t xor x t at t x x

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G. Linear and nonlinear differential equations:

An nth-order differential equation is said to be linear if it is Expressible in the form

)()()(110 xfxyxa(x)(x)ya(x)(x)ya n

)(n-(n)

Where a0(x),…., an(x) are functions of the independent variable x alone, and nonlinear otherwise. If f (x)=0, we say that (20) is homogeneous; if not, it is non-homogeneous.

(20)

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1.3 Introduction to modelingExample 1 Mechanical oscillator

afs FFFtFmx )('' (1)

))()('( AhtxFf

(2)

)(' tcxFf

)()()(')('' tFxFtcxtmx s

kxxFs )(

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Problems of Chapter 1Exercises 1.21. (b)4. (a)5. (b), (c)6. (a), (c)7.