Test 2 Review 

Covers sections 3.4, 3.5, 4.1, 5.1-5.5, 6.1-6.2

Review Problems
Try these 10 review problems - 3.4, 3.5, 4.1, 5.1, 5.2.
Try doing some of these problems again - 5.4, 5.5.
Try doing some these problems again without matlab - 6.1, 6.2.

Section 3.4 - Mechanical Vibrations 

1) Memorize the free motion of a mass on a spring equation: 

   where m is the mass of the object, c is the damping constant, F(t) = 0 (always 0 in this section) is the free force, k = spring constant. 

2) Be able to solve the spring equation by solving the characteristic equation: 

   Remember that if the roots are real and distinct, the solution is: 

     If the roots are real and repeated, the solution is: 

     If the roots are complex, r = a + bi, a - bi, the solution is: 

Section 3.5 - Nonhomogeneous Equations and Undetermined Coefficients 

1) Know the Method of Undetermined Coefficients.   

    To solve: 

   i) Solve for complimentary solution by solving the characteristic equation: 

   ii) Solve for the particular solutions by trying: 

   iii) If the trial particular solution and complimentary solution have any repeated terms, then multiply the particular solution by the lowest positive integer power of x so that no terms are repeated. 

   iv) The solution will be of the form: 

     v) Plug the solution back into the original equation and the initial conditions to solve for the undetermined coefficients. 

2) Know the Method of Variation of Parameters.   

    To solve: 

   i) Solve for complimentary solution by solving the characteristic equation: 

   ii) Set up the equations: 

   iii) Solve the equations in (ii) by substitution or elimination. 

   iv) Integrate (remember no "+ C" here). 

    v) Particular solution is then: 

Section 4.1 - First Order Systems and Applications 

1) Know how to set up first order systems as in webwork 4.1. 

2) Know how to add, subtract, scalar multiply, and multiply matrices. 

Section 5.1 - First Order Systems and Applications 

1) Know how to check a matrix solution for a differential matrix system as in problem 4 webwork 5.1 

Section 5.2 Eigenvalue Method for Homogeneous Systems 

To Solve the n x n System:                               

For Distinct Real Eigenvalues (λ's): 

1) Solve for the eigenvalues: 

2) Solve for the associated eigenvectors: 

                                *                                      

3) The general solution is then: 

           where  

For Complex Pair Eigenvalues (λ's): 

1) Solve for the eigenvalues:      

2) Solve for the associated eigenvector of λ = a + bi: 

3) Write the solution corresponding to λ: 

4) The general solution is then: 

          where  

Section 5.4 Multiple Eigenvalue Solutions 

To Solve the n x n System:                               

For Distinct Real Eigenvalues (λ's): 

1) Solve for the eigenvalues: 

2) For each eigenvalue solve for the associated eigenvectors.  For multiplicity r if you find r linearly independent eigenvectors then you can skip steps 3 and 4: 

3) For each eigenvalue of multiplicity r if you did not find r linearly independent eigenvectors in step 2, then succesively multiply this until you get the zero vector: 

   Then start with any nonzero vector and do the following: 

                                 * 

                                 * 

4) The generalized eigenvector chain is then (do not include any solutions that are a linear combination of the vectors found in step 2): 

                               * 

                               * 

5)The general solution is then: 

Section 5.5 Matrix Exponentials and Linear Systems 

For all the problems from 5.4 be able to: 

1) Set up the fundamental matrix for the system,  

2) Compute the matrix exponential,  

3) Write the solution,  

Section 6.1 - Stability and the Phase Plane 

Know all of these definitions, know how to find a critical point, know how to identify the type and stability of a critical point based on a given phase plane portrait graph: 

1) A system of the form: 

                   (1)          

is often called autonomous because the derivatives are independent of time "t". 

2) A trajectory is a parametrized solution curve, x = x(t), y = y(t), that goes through some initial point and precisely one trajectory passes through each such point. 

3) A critical point of (1) is a point (a, b) such that  

4) If (a, b) is a critical point, then 

    have 

   and therefore satisfy (1).  This solution is called an equilibrium solution. 

__________________ 

Ex.'s (Done in class) 

________________ 

We can look at the behavior of the solutions of (1) by constructing slope fields having slope: 

Such a picture is call a phase portrait or a phase plane picture. 

________________________ 

Definitions: 

1) For a critical point (a, b), if: 

    i) Either every trajectory approaches (a, b) as t goes to infinity or every trajectory recedes from (a, b) as t goes to infinity and 

    ii) Every trajectory is tangent at (a, b) to some straight line through (a, b)  

then (a, b) is called a node. 

2) A proper node has no two different pairs of "opposite trajectories" tangent to the same line; often called a "star point".  (Example below) 

3) An improper node has all trajectories tangent to a single line. (Example below) 

4) A critical point is called a sink if all trajectories approach the critical point and a source if all trajectories emanate from it.  (Both phase portraits above are sinks and if the arrows were reversed they would be sources.) 

5) A critical point is called a saddle point if only two trajectories approach the critical point, but all others are unbounded.  

(Example below) 

6) A critical point is stable if when the initial point is sufficiently close to the critical points, then the trajectory remains close to the critical point for all t > 0.  Otherwise, the critical point is called unstable.   

7) A stable center is a stable critical point surrounded by simple closed trajectories representing periodic solutions.   

(Example below) 

8) A critical point is called asymptotically stable if it is stable and every trajectory that begins sufficiently close to the critical point also approaches the critical point. 

Section 6.2 - Linear and Almost Linear Systems 

1) Memorize and be able to apply Theorem 1 to determine the type of the critical point (0, 0) of the linear systems and whether it is asymptotically stable, stable, or unstable.   

    Theorem 1 - Find the eigenvalues of the linear system with coefficient matrix A. 

      (i) If the real parts of both eigenvalues are negative, then the critical point is asymptotically stable. 

      (ii) If the real parts of both eigenvalues are zero, then the critical point is stable but not asymptotically stable. 

      (ii) If either of the real parts of the eigenvalues is positive, then the critical point is unstable. 

2) Given this chart be able to describe the point as: 

Eigenvalues of A	Type of Critical Point
Real, unequal, same sign	Improper node
Real, unequal, opposite sign	Saddle point
Real, equal	Proper or improper node
Complex conjugate	Spiral point
Pure imaginary	Center

3) Be able to find the single critical point of an almost linear systems and be apply Theorem 2 to determine the type and stability (asymptotically stable, stable, or unstable) by using the following chart. 

    Theorem 2 - Find the eigenvalues of the linear system with coefficient matrix A associated with the given almost linear system. 

      (i) If the eigenvalues are real and equal, then the critical point is either a node or a spiral point, and is asymptotically stable if they are both negative and unstable if they are both positive. 

      (ii) If the eigenvalues are both pure imaginary, then the critical point is either a center or a spiral point and may be either asymptotically stable, stable, or unstable. 

      (ii) Otherwise, classify the critical point using Theorem 1. 

Eigenvalues of A of the linearized system	Type of Critical Point of the Almost Linear System
	Stable improper node
	Stable node or spiral point
	Unstable saddle point
	Unstable node or spiral point
	Unstable improper node
	Stable spiral point
	Unstable spiral point
	Stable or unstable, center or spiral point