Applied Mathematical Sciences

I Elementary Methods for Ordinary Differential Equations of First Order.- 1. Examples and classification.- 2. Linear equations.- 3. Separable equations.- II Uniqueness and Lipschitz Conditions for Ordinary Differential Equations.- 4. First order scalar equations.- 5. Systems of equations.- 6. Higher order equations.- 7. Complex solutions.- 8. A valuable lemma.- 9. A boundary value problem.- III The Linear Equation of Order n.- 10. Constant coefficients (the homogeneous case).- 11. Linear independence and Wronskians.- 12. Constant coefficients (general solution for simple h).- 13. Variation of parameters.- IV Linear Ordinary Differential Systems.- 14. Some general properties.- 15. Constant coefficients.- 16. Oscillations and damping in applications.- 17. Variation of parameters.- 18. Matrix norm.- 19. Matrix exponential.- 20. Existence of solutions (successive approximations).- V Introduction to Delay Differential Equations.- 21. Examples and the method of steps.- 22. Some distinguishing features and some "wrong" questions.- 23. Lipschitz condition and uniqueness.- VI Existence Theory.- 24. Ordinary differential systems.- 25. Systems with bounded delays: notation and uniqueness.- 26. Systems with bounded delays: existence.- VII Linear Delay Differential Systems.- 27. Superposition.- 28. Constant coefficients.- 29. Variation of parameters.- VIII Stability.- 30. Definitions and examples.- 31. Lyapunov method for uniform stability.- 32. Asymptotic stability.- 33. Linear and quasi-linear ordinary differential systems.- 34. Linear and quasi-linear delay differential systems.- IX Autonomous Ordinary Differential Systems.- 35. Trajectories and critical points.- 36. Linear systems of second order.- 37. Critical points of quasi-linear systems of second order.- 38. Global behavior for some nonlinear examples.- Appendices.- 1. Notation for sets, functions and derivatives.- Appendices.- 2. Some theorems from calculus.- References.- Answers and Hints.