##### Winter 2014 MA212

Welcome to DE 2. To the left you'll find homework, as well as quiz and exam solutions as we take them. Below I'll be posting solutions to the daily worksheets (hopefully).

#### Matrix arithmetic

We familiarized ourselves with matrices and operations of addition, scalar multiplication, and matrix multiplication.

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#### Augmented matrices and reduced echelon form

We learned how to solve linear systems of equations by putting augmented matrices in reduced echelon form via row operations.

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#### The ``Superposition'' principle

Learned how solutions to linear homogeneous systems (so-called complimentary solutions) can be used to determine solutions to linear non-homogeneous systems. In particular, all solutions to a linear non-homogeneous system take the form of a particular solution plus the complimentary solution.

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#### Determinants and Rank

We learned about the relationship between rank and determinants.

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#### Non-singular Matrices and Their Inverses

We learned how to anticipate when a square matrix has an inverse and how to compute it.

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#### Eigenvectors and Eigenvalues

Solutions to linear systems of first order ordinary differential equations are governed by the so-called eigenvalues and eigenvectors of the associated coefficient matrix. Accordingly, we've familiarized ourselves with these terms and have gained the know-how to compute them.

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#### Separable ODEs

Certain first order ODEs can be ``separated'' in the sense that they can be written in the form f(y)y'=g(x). These ODEs are easy to solve (at least symbolically).

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#### First Order ODEs

We round out our methods for solving first order ODEs: by separation of variables, by integrating factors, by undetermined coefficients (the latter two applicable to linear first order ODEs).

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#### Linear First Order Systems of ODEs

We introduce the notion of a ``coupled'' linear system of first order ODEs.

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#### Linear First Order Systems of Homogeneous ODEs with constant coefficients: Distinct Real Eigenvalues

We determine how to derive the fundamental set of solutions to a linear system with constant coefficients in the case that the coefficient matrix has distinct real eigenvalues.

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#### Linear First Order Systems of Homogeneous ODEs with constant coefficients: Complex Eigenvalues

We determine how to derive the fundamental set of solutions to a linear system with constant coefficients in the case that the coefficient matrix has complex (conjugate) eigenvalues.

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#### Linear First Order Systems of Non-Homogeneous ODEs with constant coefficients

We determine how to derive the general solution to a linear non-homogeneous system.

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#### Mixing Problems

We apply the results of what we've learned so far to various mixing problems, namely, solvent/solute type problems and series decay.

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#### Non-Linear First Order Systems of ODEs

We determine how to find equilibrium points (constant solutions) to non-linear first order systems. We also show that changing to polar coordinates can sometimes lead to a system that is explicitly solvable.

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#### Stability in Non-Linear Systems

We determine how to classify the stability of equilibria of non-linear systems. The idea is to linearize the system and determine the stability of that linear system (by way of the Jacobian). Five times out of six, this will inform you of the stability of the non-linear system.

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#### Non-Linear Modeling: An accelerating Pendulum

We model the motion of a pendulum under constant acceleration. We determine its equilibria and classify its stability. (Use Adobe reader to view the animation).

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#### More Non-Linear Modeling: The Forever Fall

We model the motion of a particle dropped above a ``cored'' Earth. We determine and classify the stability of its equilibria.

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#### Fourier Series

We cover the basics of Fourier series; how to find coefficients and what to expect from convergence. (Use Adobe Reader to view the animation).

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#### Fourier Series Application: The Heat Equation

We model the flow of heat through a thin metal rod and derive the so-called ``heat equation''. This is a partial differential equation and Fourier series play an integral role in its solution. (Use Adobe Reader to view the animation).

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