BOOK: Differential Equation, Calculus of Variations and Special Functions

UNIT – I : Non-linear Ordinary differential Equation of Particular Forms and Riccati’s

Forms

Dr. Kamlesh Bisht

Dr. Kamlesh Bisht

(Mathematics)

Academic ConsultantDepartment of Mathematics

Uttarakhand Open University, Haldwani

Content

ObjectiveIntroductionExact non-linear Differential EquationRicatti’s EquationSolution and Application of Riccati’s EquationHomogeneous EquationReferences.

Dr. Kamlesh Bisht

(Mathematics)

ObjectivesAfter studying this unit, you should be able to-Solve exact non-linear differential equation.Solve the differential equation of the form general

Riccati’s equation.Know about the applications of the Riccati’s

equation.Solve the Riccat’s equation with one, two or three

known particular solution.Knowledge about the Homogeneous equation.

Dr. Kamlesh Bisht

(Mathematics)

Introduction In previous classes we studied a great deal about linear differential

equations of second and higher orders when coefficient may or may not be

constant.

On the other hand, the non-linear differential equations are difficult to

handle. However there is no known general methods for solving second and

higher order non linear differential equations. It is only some particular

forms that may be reduced to linear equations by suitable transformation

and integrated to yield compact result.

The aim of this unit is to study easily integrable non-linear equations.

Exact Non-Linear Differential Equations

ExampleShow that given differential equation

is an exact equation and find its solution.

Ricatti’s Equation

General Solution of Riccati’s Equation

Theorem: The Cross Ratio of any Four Particular Integrals of a Riccati’s is Independent of x

Method of solution of Riccati’s equation when particular solution is Known

Method of solution of Riccati’s equation when two particular solution are Known

Method of solution of Riccati’s equation when three particular solution are Known

Some example on Riccati’s equationQuestion: Solve

Question: Find the general solution of the Riccati’s equation whose one particular solution is (1+tanx).

222 yydx

dy

Solution of the equation of the form )(2

2

yfdx

yd

Question: Solve

Answer:

ydx

ydy cossin

2

23

Solution of the equation not containing y directly

Question: Solve

Answer: The given equation does not contain y directly. Here the lowest differential coefficient is . So Putting,

02

2

3

32

3

3

dx

yd

dx

ydx

dx

yd

2

2

dx

yd

Solution of the equation not containing x directly

Question: Solve

Solution:

Solution of the equation in which y appears in

only two derivatives whose order differ by two.

Question: Solve

Answer:

axedx

ydn

dx

yd

3

32

5

5

Solution of the equation in which y appears in only two derivatives whose order differ by unity.

Question: Solve

Answer: In the given equation y appears in two derivatives whose order differ by unity. Now substituting,

21

2

2

2

1

dx

dy

dx

yda

Homogeneous Equation

Notes:

References:Advanced Differential Equation

M.D. Raisinghania, S. Chand Publication

SLM of VMOU, Kota Differential Equation, Calculus of Variation and Special Functions.

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