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Chapter 1: First-Order Differential Equations

Chapter 1: First-Order Differential Equations. 1. Sec 1.1: Differential Equations and Mathematical Models. Definition: Differential Equation.

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Chapter 1: First-Order Differential Equations

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  1. Chapter 1: First-Order Differential Equations

  2. 1 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) 2

  3. 3 6 2 4 7 5 Sec 1.1: Differential Equations and Mathematical Models

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

  5. 8 1 Classification By Type (ODE,PDE) 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)

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

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

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

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

  10. Sec 1.1: Differential Equations and Mathematical Models 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

  11. Sec 1.1: Differential Equations and Mathematical Models Example of DE with no solution Families of Solutions What diff?????

  12. Sec 1.1: Differential Equations and Mathematical Models What diff?????

  13. Sec 1.1: Differential Equations and Mathematical Models One-parameter family of solutions What diff????? Two-parameter family of solutions

  14. Initial Value Problem Solve the IVP

  15. Page 8 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.

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

  17. Differential Equations and Mathematical Models translate Scientific laws Scientific principals Differential Equations 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.

  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 the the square root of the depth y of water in the tank is proportional to

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

  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. 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 the the square root of the depth y of water in the tank is proportional to

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