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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
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Content
ObjectiveIntroductionExact non-linear Differential EquationRicatti’s EquationSolution and Application of Riccati’s EquationHomogeneous EquationReferences.
Dr. Kamlesh Bisht
(Mathematics)
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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)
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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.
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Exact Non-Linear Differential Equations
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ExampleShow that given differential equation
is an exact equation and find its solution.
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Ricatti’s Equation
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General Solution of Riccati’s Equation
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Theorem: The Cross Ratio of any Four Particular Integrals of a Riccati’s is Independent of x
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Method of solution of Riccati’s equation when particular solution is Known
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Method of solution of Riccati’s equation when two particular solution are Known
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Method of solution of Riccati’s equation when three particular solution are Known
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Some example on Riccati’s equationQuestion: Solve
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Question: Find the general solution of the Riccati’s equation whose one particular solution is (1+tanx).
222 yydx
dy
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Solution of the equation of the form )(2
2
yfdx
yd
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Question: Solve
Answer:
ydx
ydy cossin
2
23
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Solution of the equation not containing y directly
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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
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Solution of the equation not containing x directly
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Question: Solve
Solution:
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Solution of the equation in which y appears in
only two derivatives whose order differ by two.
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Question: Solve
Answer:
axedx
ydn
dx
yd
3
32
5
5
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Solution of the equation in which y appears in only two derivatives whose order differ by unity.
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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
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Homogeneous Equation
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Notes:
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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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STAY HOME AND STAY SAFE
# COVID-19
THANKS
Dr. Kamlesh Bisht
(Mathematics) Mob. No.-
8279829875