1. Chapter 1: Introduction to Differential Equations
1.1
1.2
1.3
Definitions and Terminology
Initial Value Problem
DE as Mathematical Models

2. 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

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

4. 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

5. 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

6. Classification By Order
Write the DE
where 1 3

7. 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

8. 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

9. Classification Classification By Type Order Linearity ODE or PDE
highest derivative
Linear in y,y’,y’’,…

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

11. Navier-Stokes Equations
ODE or PDE
order??
linear or nonlinear
One Million Dollar

12. Sec 1.1: Definitions and TerminologyHW HW HW

13. 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.

14. 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

15. 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

16. 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

17. Office Hours

18. 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

19. Sec 1.1: Homework Problems

20. Sec 1.1: Homework Problems