Topics in partial differential equations

132 Pages·2019·1.663 MB·English

by  Parmanand Gupta

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ISBN 978-93-5274-102-1
9 789352 741021

TOPICS IN
PARTIAL DIFFERENTIAL
EQUATIONS
Salient features of the present edition :
(cid:2) It has detailed theory supplemented with well explained examples.
(cid:2) It has adequate number of unsolved problems of all types in exercises.
(cid:2) It has working rules for solving problems before exercises.
(cid:2) It has hints of tricky problems after relevant exercises.
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• Comprehensive Objective Mathematics (For Engg. Entrance Exam)
• Topics in Mathematics Algebra and Trigonometry (For B.A./B.Sc. I)
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(For B.A./B.Sc. I)
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• Comprehensive Differential Equations and Calculus of Variations(For B.A./B.Sc. II)
• Comprehensive Differential Equations and Differential Geometry (For B.A./B.Sc. II)
• Comprehensive Abstract Algebra (For B.A./B.Sc. III)
• Comprehensive Discrete Mathematics (For B.A./B.Sc. III, B.C.A., M.C.A.)
• Comprehensive Business Mathematics (For B.Com. I, B.T.M.)
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• A Textbook of Quantitative Techniques (For M.B.A.)
TOPICS IN
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By
PARMANAND GUPTA
B.Sc. (Hons.), M.Sc. (Delhi)
M.Phil (KU), Pre. Ph.D. (IIT Delhi)
Associate Professor of Mathematics
Former Head of Department of Mathematics
Indira Gandhi National College, Ladwa
Kurukshetra University, Haryana
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TOPICS IN PARTIAL DIFFERENTIAL EQUATIONS
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C
ONTENTS
Chapter Pages
1. Partial Differential Equations.........................................................................................1–12
1.1. Introduction................................................................................................................................. 1
1.2. Definition of a Partial Differential Equation........................................................................... 1
1.3. Order of a Partial Differential Equation .................................................................................. 1
1.4. Linear Partial Differential Equation........................................................................................ 1
1.5. Notation....................................................................................................................................... 2
1.6. Formation of a Partial Differential Equation........................................................................... 2
1.7. Formation of a Partial Differential Equation by Elimination of Arbitrary Constants......... 2
1.8. Formation of a Partial Differential Equation by Elimination of Arbitrary Functions......... 7
2. Partial Differential Equations of the First Order (Equations Linear in p and q) ....13–28
2.1. Introduction............................................................................................................................... 13
2.2. Solution of a Partial Differential Equation............................................................................ 13
2.3. Complete Solution..................................................................................................................... 13
2.4. Particular Solution ................................................................................................................... 14
2.5. Singular Solution...................................................................................................................... 14
2.6. General Solution....................................................................................................................... 14
2.7. Lagrange Linear Equation.......................................................................................................15
2.8. Solution of Lagrange Linear Equation ................................................................................... 15
3. Partial Differential Equations of the First Order (Equations Non-linear in
p and q) ..........................................................................................................................29–59
3.1. Introduction............................................................................................................................... 29
3.2. Special Type I : Equations Containing Only p and q.............................................................29
3.3. Special Type II : Equations of the Form z = px + qy + g(p, q)............................................... 34
3.4. Special Type III : Equations Containing Only z, p and q...................................................... 39
3.5. Special Type IV : Equations of the Form f (x, p) = f (y, q)..................................................... 44
1 2
3.6. Use of Transformations............................................................................................................50
3.7. Charpit’s General Method of Solution ....................................................................................53
4. Homogeneous Linear Partial Differential Equations with Constant Coefficients .60–80
4.1. Introduction............................................................................................................................... 60
4.2. Partial Differential Equations of Second and Higher Order ................................................ 60
4.3. Homogeneous Linear Partial Differential Equations with Constant Coefficients..............60
4.4. Some Theorems......................................................................................................................... 61
4.5. General Solution of Homogeneous Linear Partial Differential Equation f(D, D′)z = 0
with Constant Coefficients.......................................................................................................62
(v)
Chapter Pages
4.6. General Solution of Homogeneous Linear Partial Differential Equation
f(D, D′)z = F(x, y) with Constant Coefficients.........................................................................66
4.7. Particular Integral of f(D, D′)z = F(x, y).................................................................................. 66
4.8. Particular Integral When F(X, Y) is Sum or Difference of Terms of the Form xmyn........... 66
4.9. Particular Integral When F(x, y) is of the Form φ(ax + by)................................................... 68
4.10. General Method of Finding Particular Integral..................................................................... 75
5. Non-homogeneous Linear Partial Differential Equations with Constant
Coefficients....................................................................................................................81–97
5.1. Introduction............................................................................................................................... 81
5.2. Non-homogeneous Linear Partial Differential Equations with Constant Coefficients..... 81
5.3. Reducible and Irreducible Non-homogeneous Linear Partial Differential Equations with
Constant Coefficients................................................................................................................ 81
5.4. General Solution of Reducible Non-homogeneous Linear Partial Differential Equation
f(D, D′)z = 0 with Constant Coefficients ................................................................................. 82
5.5. General Solution of Irreducible Non-homogeneous Linear Partial Differential Equation
f(D, D′)z = 0 With Constant Coefficients................................................................................. 85
5.6. General Solution of Non-homogeneous Linear Partial Differential Equation With
Constant Coefficients................................................................................................................ 88
5.7. Particular Integral of f(D, D′)z = F(x, y).................................................................................. 88
5.8. Particular Integral When F(x, y) is Sum or Difference of Terms of the Form xmyn............ 88
5.9. Particular Integral When F(x, y) is of the Form eax+by........................................................... 91
5.10. Particular Integral When F(x, y) is of the Form sin (ax + by) or cos(ax + by)...................... 93
5.11. Particular Integral When F(x, y) is of the Form eax+by V(x, y)............................................... 95
6. Partial Differential Equations Reducible to Equations with
Constant Coefficients.................................................................................................98–103
6.1. Introduction............................................................................................................................... 98
6.2. Reducible Linear Partial Differential Equations with Variable Coefficients...................... 98
6.3. Solution of Reducible Linear Partial Differential Equations with Variable Coefficients..98
7. Monge’s Methods......................................................................................................104–118
7.1. Introduction.............................................................................................................................104
7.2. Partial Differential Equation of Second Order ....................................................................104
7.3. Intermediate Integral.............................................................................................................104
7.4. Monge’s Methods..................................................................................................................... 104
7.5. Monge’s Method of Solving Rr + Ss + Tt = V........................................................................105
7.6. Monge’s Method of Solving Rr + Ss + Tt + U(rt – s2) = V....................................................113
(vi)
P
REFACE
The present book on ‘‘Partial Differential Equations’’ has been written as a textbook
according to the latest guidelines and syllabus in Mathematics issued by the U.G.C. for various
universities. The text of the book has been prepared with the following salient features:
(i)The language of the book is simple and easy to understand.
(ii)Each topic has been presented in a systematic, simple, lucid and exhaustive
manner.
(iii)A large number of important solved examples properly selected from the previous
university question papers have been provided to enable the students to have a clear
grasp of the subject and to equip them for attempting problems in the university
examination without any difficulty.
(iv)Apart from providing a large number of examples, different type of questions in am-
ple quantity have been provided for a thorough practice to the students.
(v)A large number of ‘notes’ and ‘remarks’ have been added for better understanding of
the subject.
A serious effort has been made to keep the book free from mistakes and errors.
In fact no pains have been spared to make the book interesting and useful.
Suggestions and comments for further improvement of the book will be welcomed.
—AUTHOR
(vii)