Differential Equations and Linear Algebra.

Artur Jackson
Office: MATH 701
Mailbox: MATH 835

2014 Fall (CURRENT)

Quizzes:

2013 Spring (OLD)

Quizzes:

Notes:

1. When using the existence and uniqueness theorem why is it imporant that the bounding rectangle contain more than one point. To answer this, try coming up with a function \(f:\mathbb{R}^2\rightarrow \mathbb{R}\) which is continuous only at the origin. (What can you say about solutions, if any exist, to the equation \(\frac{dy}{dx} = f(x,y)\)?)

   Existence and uniqueness is not just of purely theoretical interest. It is of important interest in controls systems, electrical engineering, physics, nuclear engineering, etc. For example, suppose one determines the equations governing behavior of a system and one manages to find a solution. It is important to understand whether some other types of behavior can be ruled out. This is the case when the existence and uniqueness theorem applies.

2. Integrating factors \(\mu=\mu(x)\) are useful for solving first order equations \(\frac{dy}{dx} + p(x)y = q(x)\). But what property of the integrating factor allows to claim that the solution spaces of the original equation and the modified equation \(\mu \frac{dy}{dx} + \mu p(x) = \mu q(x)\) are identifcal?

   Can you find a function \(\mu=\mu(x)\) (not necessarily an integrating factor) and an equation \(\frac{dy}{dx} + p(x)y = q(x)\) such that the modified equation \(\mu \frac{dy}{dx} + \mu p(x) = \mu q(x)\) has different solutions than the original equation?