Chapter 1: Introduction to Differential Equations 1.1 1.2 1.3 Definitions and Terminology Initial Value Problem DE as Mathematical Models
Sec 1.1: Definitions and Terminology Definition 1.1 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) 6 1 3 2 4 7 5
Classification 8 9 Classification By Type Order Linearity Sec 1.1: Definitions and Terminology Classification 8 9 Classification By Linearity 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
Classification By Order Write the DE where 1 3
Classification By Linearity An n-th order ODE Is said to be linear If F is linear in This means that F = The power of y,y’,y’’,…. Is 1 2) The coefficients depend on x only. 3) No nonlinear function in y 1 2
Classification By Linearity The power of y,y’,y’’,…. Is 1 2) The coefficients depend on x only. 3) No nonlinear function in y 1 2 5 3 6 4
Classification Classification By Type Order Linearity ODE or PDE highest derivative Linear in y,y’,y’’,…
System of DE System of two Ordinary Differential Equations 2ed order, linear, ODE
Navier-Stokes Equations ODE or PDE order?? linear or nonlinear One Million Dollar
Sec 1.1: Definitions and Terminology HW HW HW
Sec 1.1: Definitions and Terminology Definition 1.2 Solution of an ODE Any function defined on an interval I and possessing at least n derivatives that are continuous on I, which when substituted into an nth-order ODE reduces the equation to an identity, is said to be a solution of the equation on the interval.
Sec 1.1: Definitions and Terminology Definition 1.2 Solution of an ODE Any function defined on an interval I and possessing at least n derivatives that are continuous on I, which when substituted into an nth-order ODE reduces the equation to an identity, is said to be a solution of the equation on the interval. Interval of existence = interval of validity = domain of the solution
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
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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
Sec 1.1: Homework Problems
Sec 1.1: Homework Problems