Lecture 6: Let’s Get Visual

Name:

Lecture Notes-

Direction Fields (Slope Fields) & Integral Curves-Graphical Methods

Definitions:

1. Direction (Slope) Field-

2. Isoclines-

3. Integral Curves-

Example:

4.

Existence and Uniqueness Theorem

Criteria for Existence and Uniqueness Theorem:

For the Initial Value Problem (I.V.P.):

1.

2.

Examples:

3.

4.

5.

Squeeze Theorem:

In calculus, the squeeze theorem (known also as the pinching theorem, the sandwich theorem, the sandwich rule and sometimes the squeeze lemma) is a theorem regarding the limit of a function.The squeeze theorem is a technical result which is very important in proofs in calculus and mathematical analysis. It is typically used to confirm the limit of a function via comparison with two other functions whose limits are known or easily computed.

Mathematical Definition:

Example:

1. What is the limit of ?

Squeeze Theorem & Direction Fields

2.

3.

Euler’s Method-Numerical Methods

The graphical method gives you a quick feel for how the integral curves behaves. But when they must be known accurately and the equation cannot be solved explicitly, numerical methods are used. The simplest numerical method is Euler’s method.

Example:

For the I.V.P: use Euler’s method with a step size of to find.

n / Xn / Yn / F(Xn,Yn) / h* F(Xn,Yn)
0
1
2

Remarks:

1.

2.

3.

4.

The Second Derivative & Concavity:

Assume . Use Euler’s method to estimate . Is the estimate too high or too low? Explain.

Autonomous DE’s

In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not depend on the independent variable.

Mathematical Definition:

Characteristics of Autonomous DE’s:

1.

2.

3.

4.

Analysis of Autonomous DE’s:

5.

6.

7.

8.

9. Phase Line-

Examples:

10. Newton’s Law of Cooling:

11. Logistic DE:

Bifurcation Diagrams

As we’ve seen the dynamics of vector fields on the line is very limited; all solutions either settle down to equilibrium or head out to ∞

Given the triviality of the dynamics, what’s so interesting about studying one-dimensional systems? Answer: Dependence on parameters.The qualitative structure of flow can change as parameters are varied. In particular, fixed points can be created or destroyed, or their stability can change. These qualitative changes in dynamics are called bifurcations, and the parameter values at which they occur are called bifurcation points.

Bifurcations are important scientifically in that they provide models of transitions and instabilities as some control parameter is varied.

Physical Example:

Example:

1. Logistic DE with Harvesting:

When the graph of is a parabola. Note: As h changes the parabola moves up and down.