1. Chapter 1: First-Order Differential Equations 1

2. Sec 1.1: Differential Equations and Mathematical Models Definition: Differential Equation An equation containing the derivatives of one or more dependent variables with respect to one or more independent variables, is said to be a differential equation (DE) 1 2 2

3. Sec 1.1: Differential Equations and Mathematical Models 2 3 4 5 67 3

4. Sec 1.1: Definitions and Terminology 8 9 Classification Classification By Order Type 4

5. Classification By Type If an equation containing only ordinary derivatives  it is said to be Ordinary Differential Equation (ODE) An equation involving partial derivatives  it is said to be Partial Differential Equation (PDE) 8 1 Classification By Type (ODE,PDE) 5

6. Classification By Order The order of a differential equation (ODE or PDE) is the order of the highest derivative highest derivative in the equation. 1 2 3 4 5 Classification By Order. n-th order DE 6

7. highest derivative ODE or PDE Classification By Order Type Classification 7

8. System of DE System of two Ordinary Differential Equations 2ed order, linear, ODE 8

9. Navier-Stokes Equations  ODE or PDE  order?? One Million Dollar 9

10. Definition: Solution of an ODE A continuous function is said to be a solution of a DE if it satisfies the DE on an interval I Sec 1.1: Differential Equations and Mathematical Models 10

11. Families of Solutions Sec 1.1: Differential Equations and Mathematical Models 11

12. Sec 1.1: Differential Equations and Mathematical Models 12

13. Sec 1.1: Differential Equations and Mathematical Models One- paramet er family of solutions Two- paramet er family of solutions 13

14. 14 Initial Value Problem Solve the IVP

15. 15 The central question of greatest immediate interest to us is this : if we ate given a differential equation known to have a solution satisfying a given initial condition. How do we actually find or compute that solution? And, once found, what can we do with it? We will see that a relatively few simple technique  Separation of variables 1.4  Solution of linear equations 1.5  Elementary substitution method 1.6 Are enough to enable us to solve a varity of first-order equations having impressive applications. The central question of greatest immediate interest to us is this : if we ate given a differential equation known to have a solution satisfying a given initial condition. How do we actually find or compute that solution? And, once found, what can we do with it? We will see that a relatively few simple technique  Separation of variables 1.4  Solution of linear equations 1.5  Elementary substitution method 1.6 Are enough to enable us to solve a varity of first-order equations having impressive applications. Page 8

16. 16 The study of DE has 3 principal goals 1) To discover the differential equation that describes a specified physical situation 2) To find either exactly or approximately the appropriate solution of that equation 3) To interpret the solution that is found Find DE Find sol interpret sol

17. 17 Differential Equations and Mathematical Models Torricelli’s law implies: The time rate of change of the volume V(t) of water in a draining tank is propotional to the square root of the depth y of water in the tank. Scientific laws Scientific principals Scientific laws Scientific principals Differential Equations Differential Equations translate

18. 18 Differential Equations and Mathematical Models Torricelli’s law implies: The time rate of change of the volume V(t) of water in a draining tank is propotional to the square root of the depth y of water in the tank. The The time rate of change of the volume V(t) of water in a draining is proportional to the the square root of the depth y of water in the tank

19. 19 Differential Equations and Mathematical Models Scientific laws Scientific principals Scientific laws Scientific principals Differential Equations Differential Equations translate Newton’s law of cooling: The time rate of change of the temperature T(t) of a body is proportional to the difference between T and the temperature A of surrounding medium

20. 20 Differential Equations and Mathematical Models The time rate of change of the temperature T(t) of a body is proportional to the difference between T and the temperature A of surrounding medium

21. 21 Differential Equations and Mathematical Models Torricelli’s law implies: The time rate of change of the volume V(t) of water in a draining tank is propotional to the square root of the depth y of water in the tank.

22. 22 Differential Equations and Mathematical Models Torricelli’s law implies: The time rate of change of the volume V(t) of water in a draining tank is propotional to the square root of the depth y of water in the tank. The The time rate of change of the volume V(t) of water in a draining is proportional to the the square root of the depth y of water in the tank

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