Linear Sys. w/ Const. Coeff.

Due: Fri. 14-Mar

Due: Tue. 18-Mar

Due: Fri. 21-Mar

Due: Thur. 27-Mar

Due: Fri. 4-Apr.

Due: Wed. 9-Apr.

Due: Fri. 11-Apr.

Due: Tue. 15-Apr.

Due: Fri. 2-May.

Due: Tue. 6-May.

Due: Fri. 16-May.

##### 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.

#### Day 1: Matrix arithmetic

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

###### Worksheet - .pdf

#### Day 2: 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.

###### Worksheet - .pdf

#### Day 3: Homogeneous systems and matrix equations

We discussed the general theory of homogeneous systems of linear equations, and matrix equation forms of systems of equations in general.

###### Worksheet - .pdf

#### Day 4: Rank

We introduced the notion of matrix rank and discussed how it relates to the solvability of matrix equations. We also took our first quiz.

#### Day 5: Rank and determinants

We related the notion of matrix rank to the notion of matrix determinant. The main point is that for n-by-n matrices A, the rank of A is n iff the determinant of A is non-zero.

###### Worksheet - .pdf

#### Day 6: Singular and non-singular matrices

To solve Ax = b, one might hope to multiply both sides of this equality by the inverse of A. Accordingly, we defined matrix inverses, when to expect them to exist, and how to compute them (if they exist).

###### Worksheet - .pdf

#### Day 7: MAPLE

We spent the day playing around with MAPLE (how to input a matrix, matrix operations, solving systems of linear equations, etc).

###### Worksheet - .pdf

#### Day 8: Eigenvalues and Eigenvectors

We introduced the notion of eigenvectors and eigenvalues. For 2x2 matrices A with distinct eigenvalues l and m with associated eigenvectors v and w (respectively), we noticed the following nice geometric interpretation of the situation: for an arbitrary vector x, Ax is the vector x stretched in the v-ward direction by l and stretched in th w-ward direction by m. So knowing the eigenvalues/eigenvectors of a matrix generally puts you in a better position to anticipate the value of Ax.

#### Day 9: Eigenvalues and Eigenvectors Cont.

We tied together our knowledge of rank, singularity, and eigenvalue.

###### Worksheet - .pdf

#### Day 10: Complex Arithmetic

We reviewed the basics of complex arithmetic paying special attention to the complex exponential and the role it plays in computing powers and roots of complex numbers.

###### Worksheet - .pdf

#### Day 14: Separable DE's and Integrating factors

We reviewed some of the basic methods for solving ODE's, namely the method of separating variables and integrating factors.

###### Worksheet - .pdf

#### Day 15: Linear Systems of DE's

We introduced systems of linear differential equations and the basic terminology we'll need going forward.

###### Worksheet - .pdf

#### Day 16: Linear Systems of DE's continued

We continued our introduction into systems of linear DE's and took our third quiz.

#### Day 17: Homogeneous Systems of Linear DE's

We saw how to construct solutions to homogeneous systems of linear DE's from the eigenvalues and eigenvectors of the matrix of coefficients of those systems.

###### Worksheet - .pdf

#### Day 22: Non-homogeneous Systems of Linear DE's

In order to solve non-homogeneous systems, we employ something of an educated guess-and-check method known as undetermined coefficients.

###### Worksheet - .pdf

#### Day 26: Non-Linear Autonomous Systems

We introduced non-linear autonomous systems of differential equations and began to describe their phase portraits and equilibrium points. You may find the the following Java

applet useful for plotting phase portraits of autonomous systems in addition to MAPLE's capabilities.

###### Worksheet - .pdf

#### Day 27: Stability of Nonlinear Autonomous Systems

We defined the notions of stable, asymptotically stable, and unstable equilibrium solutions to nonlinear autonomous systems. We also had our fifth quiz.

#### Day 29: Classifying Stability in Linear Autonomous Systems

In a manner analogous to the approximation of linear functions to differential functions, we described how linear systems approximate non-linear autonomous systems (with differentiable components). This approximation allows us to use what we know about stability of linear systems to describe stability in non-linear autonomous systems. Accordingly, we completely characterized the stability of linear systems from the trace and determinant of the associated matrix.

###### Worksheet - .pdf

#### Day 30: Classifying Stability in Nonlinear Autonomous Systems

We discussed what information can be gathered about the stability of equilibrium solutions to non-linear autonomous systems from their linearizations. We also discussed other methods for analysing the stability of non-linear systems, namely, the so-called phase-plane method.

###### Worksheet - .pdf