CSIR NET EXAM – ORDINARY DIFFERENTIAL EQUATION NOTES

Complete set of notes on LINEAR ALGEBRA for CSIR NET exam that is exclusively compiled & managed by Dr Gajendra Purohit.

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Table of Content Order of differential equation Degree of differential equation Linear differential equation Non Linear differential equation Equation of first order and first degree Exact differential equation Linear differential equation with constant coefficients Auxillary equation Linear differential equation with variable coefficients UNIQUENESS AND EXISTENCE OF FOFD The Lipschitz Condition

 

Definition :

An equation involving derivatives with respect to an independent variable and involve dependent variable is called an Ordinary Differential Equation.

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Order of Differential Equation :

The order of highest derivatives is called the order of Differential Equation.

Example :

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Degree of Differential Equation :

The degree of the highest derivatives is called degree of the Differential Equation.

Example :

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Linear differential equation :

The Differential Equation is called linear if

(i) every dependent variable and its derivatives occurs in the first degree only

(ii) No product of dependent variable and its derivatives.

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Non-linear Differential Equation :

The Differential Equation which is not linear is called non-linear Differential Equation.

EQUATIONS OF FIRST ORDER AND FIRST DEGREE

A General differential equation of first order and first degree is of the form

where M, N are functions of x and y both.

CaseI:  Variable separable:

An equation whose variables are separable and can be put in the form g(x)dx+h(y)dy=0 is called an equation of variable separable form. Integrating, ∫g(x)dx+∫h(y)dy=0 , where c is an arbitrary constant. This is the general solution of the differential equation.

Case II: Equations reducible to variable separable: To solve the equation

Method to solve:

Putting in

which is of the type “Variable Separable” and hence can be solved.

Case III: Equations reducible to homogeneous equations

To solve the equation

Method to solve:

• Put x = X +h , y = Y + k.
• Equate the constant terms of numerator and denominator to zero and find the values of h and k.
• Proceed as in case III

Method to solve:

• Put a₁ x + b₁ y = v
• The equation gets reduced to variable separable form in v and x.

Case IV:  Linear differential equations of first order

Case V: Equations reducible to linear differential equation

Method to solve

• Dividing throughout by yⁿ.
• Putting y^1-n = z and get the linear equation in z.
• Proceed as in Case V.

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Exact Differential Equation

Definition

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Linear differential equation with constant coefficients

Definition:

A linear differential equation with constant coefficients is one in which the dependent variable and its differential coefficients occur only in the first degree and are not multiplied together.

(Where P₀, P₁, P₂, …,P_nare constants and Q is a function of x) is the linear equation of the n^th order.

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Auxiliary Equation (A.E.):

Auxiliary equation is obtained by equating to zero the symbolic co-efficients of y. Thus,

Method to solve the equation:

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Linear differential equations with variable coefficients

Definition:

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UNIQUENESS AND EXISTENCE OF FOFD

Then the necessary and sufficient condition such that. . has a solution

Iff  f(x, y) is integrable.

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The Lipschitz Condition :

Theorem :

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Existence and Uniqueness theorem :

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