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Chapter 1: First-Order Differential Equations
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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
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Sec 1.1: Differential Equations and Mathematical Models
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Classification 8 9 Classification By Type Order
Sec 1.1: Definitions and Terminology Classification 8 9 Classification By Order Type
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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
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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
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Classification Classification By Type Order ODE or PDE
highest derivative Type Order
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System of DE System of two Ordinary Differential Equations
2ed order, linear, ODE
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Navier-Stokes Equations
ODE or PDE order?? One Million Dollar
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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
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Sec 1.1: Definitions and Terminology
Families of Solutions What diff?????
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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
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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
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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
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Sec 1.1: Homework Problems
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Sec 1.1: Homework Problems
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