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The present book Differential equations provides a detailed account of the equations of First Order and First Degree, Singular solutions and Orthogonal Trajectories, Linear differential equations with constant coefficients and other miscellaneous differential equations.
The numerous high-graded solved examples provided in the book have been mainly taken from the authoritative textbooks and question papers of various university and competitive examinations which will facilitate easy understanding of the various skills necessary in solving the problems. In addition, these examples will acquaint the readers with the type of questions usually set at the examinations. Furthermore, practice exercises of multiple varieties have also been given, believing that they will help in quick revision and in gaining confidence in the understanding of the subject. Answers to these questions have been verified thoroughly. It is hoped that a through study of this book would enable the students of Mathematics to secure high marks in the examinations. Besides students, the teachers of the subject would also find it useful in elucidating concepts to the students by following a number of possible tracks suggested in the book.
Contents: Preface. I. Differential equations of the First Order and the First Degree:
1. Definitions.
2. Arbitrary Constants.
3. Equations of the First Order and First Degree.
4. Equations in which the variables are separable.
5. Homogeneous function.
6. A test for homogeneity of a function of X and Y.
7. Homogeneous equations.
8. Equations reducible to a homogeneous form.
9. Linear differential equations.
10. Equations reducible to linear form.
11. Exact differential equations.
12. Condition that an equation of First Order and First Degree be exact.
13. Solution of an exact differential equation.
14. Integrating factor.
15. Integrating factor by inspection.
16. Rules for finding out integrating factors.
17. Change of variables. II. Different equations of the First Order, but not of the First Degree:
18. Types of such equations.
19. Equations solvable for p.
20. Equations solvable for Y.
21. Equations solvable for X.
22. Clairaut's equation. III. Singular solutions and Orthogonal Trajectories:
23. Singular solution.
24. Determination of Singular solution.
25. Particular case.
26. Trajectories.
27. Orthogonal Trajectory.
28. Differential equation of Orthogonal Trajectories. IV. Linear differential equations with constant coefficients:
29. Definition.
30. Theorem 1.
31. Theorem 2.
32. Theorem 3.
33. Complementary function.
34. The symbol D.
35. Auxiliary equation having equal roots.
36. Auxiliary equation having imaginary roots.
37. The particular integral.
38. Shorter methods of finding particular integrals in certain special cases. V. Miscellaneous differential equations:
39. Homogeneous linear equations.
40. Equations reducible to homogeneous linear form.
41. Simultaneously linear differential equations with constant coefficients