##### 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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#### Systems of Linear Equations

We learned that systems of linear equations can be easily solved by putting augmented matrices into reduced echelon form.

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#### Determinants and Rank

We learned that the consistency or solvability of a system is influenced by the determinant and rank of the matrix of coefficients.

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#### Singular and Invertible Matrices

We learned that if a matrix is non-singular, then it can be ``undone'' or inverted. This is handy for solving systems like Ax=b for multiple different b. We also related these notions to those of determinant, rank, and consistency of linear systems

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#### Eigenvalues and Eigenvectors

Given a system of differential equations x' = Ax, we noticed that solutions were determined by solutions to Av = (lambda)v. This introduced us to the very important notion of eigenvalue (lambda) and eigenvector (v).

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#### Separable ODEs

We learned how to solve first order ``separable'' ODEs.

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#### Other methods for solving first order ODEs

We learned how to solve linear first order equations by integrating factors and undetermined coefficients.

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#### Linear Systems of ODEs

We introduced linear systems of ODEs and discussed what to expect from their solutions.

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#### Homogeneous Systems of ODEs with constant coefficients: distinct real eigenvalues

We discussed how to solve a 2x2 system where the matrix of coefficients has two distinct eigenvalues.

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#### Homogeneous Systems of ODEs with constant coefficients: complex eigenvalues

We discussed how to solve a 2x2 system where the matrix of coefficients has complex conjugate eigenvalues.

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#### Homogeneous Systems of ODEs with constant coefficients: repeat real eigenvalues

We discussed how to solve a 2x2 system where the matrix of coefficients has a repeat real eigenvalue.

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#### Non-Homogeneous Systems of ODEs with constant coefficients

We discussed how to solve non-homogeneous systems by the method of undetermined coefficients.

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#### Mixing Problems

We worked through a couple mixing problems (salt tank and series decay).

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#### Non-Linear Systems of Differential Equations

We discussed equilibrium points of non-linear system and paid lip-service to the process of changing variables (cartesian to polar).

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#### Classifying Equilibria on Non-Linear Systems: Via Jacobians

We discussed how to use the Jacobian (or the linearization) of a non-linear system to help classify its equilibria.

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

In class, we discussed how to model an undamped pendulum. We go a little farther in this worksheet and assume the pendulum is accelerating (for instance if it was placed in an accelerating car) and under the influence of a damping force.

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

In this example, we model the motion of a particle falling through a ``cored'' planet.

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#### Phase-Plane Method

These examples illustrate the phase-plane method. It would be a good idea to attempt the problem before reading the solution.

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#### Heat Equation

We discussed the 1-dimensional Heat Equation.

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#### Fourier Series

We discussed how to compute Fourier Series.

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