DIFFERENTIAL EQUATIONS INTRODUCTION

Differential Equation:
an equation containing one or more derivatives of an unknown function.

ODE's and PDE's

1. Ordinary Differential Equations
   involve only ordinary derivatives.
2. Partial Differential Equations
   involve at least one partial derivative.
3. A system of DE's is a set of one or more simultaneous ODE's and/or PDE's involving more than one unknown function.
4. The order of an ODE, PDE or system of DE's is
   the order of the highest derivative.

Linear and Nonlinear Equations

1. A DE is linear if the unknown function(s) and derivatives are combined linearly to form the DE.
2. A DE is nonlinear if it is not a linear DE.

Solutions to DE's

1. A solution to a DE is a function that satisfies the DE.
2. Issues:
   1. What is the domain for the solution?
   2. Is there always a solution?
   3. Is the solution unique? Initial Values and Integral Curves?
   4. How can the solution be determined? Formula? Computer?
3. Central problem for Math 315: given a DE, find the solution(s).

DIFFERENTIAL EQUATIONS INTRO CONTINUED

Computer Methods

1. Symbolic Methods
2. Numerical Methods
3. Graphics
   1. Plot solutions
   2. Plot functions of solutions
   3. Qualitative Analysis
      Direction Fields: plots of tangent lines for solutions

Scope of Math 315

1. First Order DE's: linear and nonlinear (3 weeks)
2. Numerical Methods (1 week)
3. Linear Second Order Equations (3 weeks)
4. Linear Higher Order Equations (1 week)
5. Series Solutions to Second Order Equations (1 week)
6. Laplace Transforms (2 weeks)
7. Systems of First Order Linear Equations (2 weeks)

Simple Problems

• Population Growth for population :
  1. $p' = rp$, with growth rate $r$ ;
  2. $p' = rp - k$, growth rate $r$, constant kill factor $k$;

• Falling Object with velocity $v(t)$:
  1. $mv' = mg$ , without air resistance;
  2. $mv' = mg - \gamma v$, with air resistance proportional to .