Chapter 1: First-Order Differential Equations

Sec 1.1: Differential Equations and Mathematical Models Definition: Differential Equation 1 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

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

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

Classification By Type 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) 8 1

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

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

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

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

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

Sec 1.1: Definitions and Terminology Families of Solutions What diff?????

Sec 1.1: Definitions and Terminology Families of Solutions One-parameter family of solutions Two-parameter family of solutions n-parameter family of solutions Particular Solution: A solution of DE that is free of arbitrary parameters

Sec 1.1: Definitions and Terminology Trivial Solution y=0 is said to be trivial solution Singular Solution sometimes a DE has a solution that is not a member of a family of solutions. This solution is called singular solution one-parameter family of solutions Verify?? Not a member of the family no value for c

Piecewise-Defined Solution Explicit and Impicit Solutions Sec 1.1: Definitions and Terminology Piecewise-Defined Solution Verify?? Explicit and Impicit Solutions Explicit Solution Verify?? Implicit Solution

Sec 1.1: Homework Problems

Sec 1.1: Homework Problems