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CALPOLY / Math / Mat 244 / Name what are the two levels of DE.

Name what are the two levels of DE.

Name what are the two levels of DE.

Description

School: California Polytechnic State University - San Luis Obispo
Department: Math
Course: Linear Analysis
Professor: Nash
Term: Spring 2016
Tags: Math, math244, cal poly, Nash, Linear, analysis, math notes, and linear notes
Cost: 25
Name: Math 244, Week 1, Notes
Description: Lecture notes covering week 1 of the course
Uploaded: 04/04/2016
9 Pages 171 Views 0 Unlocks
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Math 244If you want to learn more check out cal poly ce

1.2 Differential Equations (DE)

  • An equation relating same quantity and its derivative(s)

        Don't forget about the age old question of What is autonomous consumption?

If you want to learn more check out What does crowding out mean?

Solutions to DE’s are functions

         has a solution y = We also discuss several other topics like What is a super antioxidant vitamin?

2 Levels of DE

1) General Solutions : (Infinite) families of all solutions to a DE…

  • Parametrized by a constant (e.g. C)
  • For Example :

If you want to learn more check out How do you represent in Python?

2) Particular Solution: A specific function that satisfies the DE and an initial condition

  • E.g.
  • Finding a particular solution is also called solving an initial value problem
  • For Example         {

{        so We also discuss several other topics like What is the formula for finding longitude stress in beams?

1.2 + 1.4 Classification of DE

*Order (# of derivatives) matters*

Def

Order of a DE is the order of the highest derivative which appears in the DE.

Linear Differential Equations: A DE is called linear if it can be written as a degree 1 polynomial in the “variables”

        E.g.  is linear

And

        

1st Order DE Types

Pure-Time DE:

  • Solve by integrating

Autonomous DE:

E.g.        

             or

        any constant function which makes the right side = 0 is a solution.

1.4        Separable DE

Def:

A (1st order) DE is separable if it can be written in the form:

        

Solving a Separable DE

  • Separating variables (x’s on one side, y’s on the other)
  • E.g. 
  • Integrate both sides
  • Isolate y (ater integration)

Ex. Solve:        

                

                

                

                

                

1.3 Direction FIelds (Slope FIelds)

  • Solutions to non separable differential equations can be understood graphically
  • For Instance:

Consider the equation

fix various initial conditions (values of x and y) ; y(a)=b

Plot the corresponding slope  at the point (a,b)

Ex.                                         

x

y

0

0

0

0

1

1

0

2

2

0

-1

-1

1

0

-1

2

0

-2

-1

0

1

                Trends

x

y

n

n

o

n+1

n

-1

n

n+1

1

Special Features

Isoclines: curves in xy plane along which  stays constant

        E.g. y=x is an isocline for D.E

Equilibrium Solutions:

  • Asymptotes to the slope field
  • Constant solutions y=r
  • Identified by finding the y-value roots of the right hand side
  • Graphically equilibrium solutions are horizontal lines.

E.g.

                Has y=0 and y=1 as equilibrium solutions

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