Things to know - Test #2
(sections 4.5, 4.6, 8.1, 8.2)
MA3521, Fall '03, T. Olson

Note: On the test, you may use your calculator to find derivatives and antiderivatives(integrals), to compute eigenvalues and eigenvectors, and to do row reduction of matrices. You should show all other work by hand.

Know these terms and relationships between them:

• differential operator
• annihilate, annihilator
• system of D.E.s
• linear system
• matrix form of linear system
• homogeneous system
• general solution (for system)
• particular solution (for system)
• fundamental solution set
• linear dependence/independence
• Wronskian (for system)
• eigenvalue/eigenvector
• multiplicity of eigenvalue
• undetermined coefficients
  (annihilator method for y_p)
• variation of parameters
  for second-order equations
  (assume y_p=u₁(x) y₁(x) + u₂(x) y₂(x))
  
  (From Test # 1)
• solution (to a differential equation)
  • family of solutions
  • general solution
  • particular solution
• auxiliary equation
  (for homog. linear const.-coeff.  eq.)
• initial-value problem
• boundary-value problem
• order (of a differential equation)
• homogeneous, non-homogeneous
• complementary function

Things to think about:

1.

What is the ``annihilator method'' (undetermined coefficients)? What kinds of equations can it be used with, and what type of solution does it provide?

2.

What differential operator annihilates polynomials? ...exponentials? ...sines and cosines? ...products of exponentials with sines/cosines? ...products of x or x² with any of the preceding items? ...linear combinations of these things?

3.

When using the annihilator method to find a particular solution to a linear D.E. with constant coefficients, why do you bother to find the solution to the homogeneous equation first? (How does this affect the form you assume for your particular solution?)

4.

Once you've used the annihilator method to determine the form of a particular solution, how do you compute y_p?

5.

Once you've got a particular solution to a D.E., how do you write the general solution?

6.

What kinds of equations can you solve using variation of parameters (i.e., y_p=u₁ y₁ + u₂ y₂)? What kind of solution does it help you find?

7.

In a solution by variation of parameters
( y_p(x)=u₁(x) y₁(x) + u₂(x) y₂(x)), what are y₁ and y₂? What equations must u₁ and u₂ satisfy?

8.

What does a LINEAR system of differential equations look like?

9.

Given a linear system of D.E.s, how do you write it in matrix form, and vice-versa?

1.

What is the general matrix form for a linear, first-order differential equation if it's homogeneous? ...if it's not homogeneous?

2.

How do you check if a vector of functions is a solution to a linear system of first-order D.E.s?

3.

Given a set of solutions to a homogeneous linear system, how do you check whether they are linearly independent? How do you decide if they form a fundamental solution set?

4.

If you have a fundamental set of solutions to a homogeneous linear system, how do you form the general solution? How many arbitrary constants are involved?

5.

For linear systems, how is the general solution to the non-homogeneous equation related to the solution to the homogeneous equation? How many arbitrary constants are involved?

The remaining ask about solving systems of the form $\vec{X}' =A\vec{X}$, where A is constant and $\vec{X}=\vec{X}(t)$.

6.

What kinds of solutions do you have when the eigenvalues of A are

(a)

real and distinct?

(b)

real and distinct and negative?

(c)

real, but with some repeated eigenvalues? (two cases here)

(d)

imaginary?

(e)

complex?

(f)

complex, with negative real part?

7.

Suppose $\lambda$ is a real eigenvalue of A with eigenvector $\vec{K}$. What is one solution to $\vec{X}' =A\vec{X}$?

8.

Suppose $\lambda$ is a real eigenvalue of A with a multiplicity of 2 and 2 distinct eigenvectors, $\vec{K}_1$ and $\vec{K}_2$. What are two solutions to $\vec{X}' =A\vec{X}$?

9.

Suppose $\lambda$ is a real eigenvalue of A with a multiplicity of 2, but with only one eigenvector, $\vec{K}_1$. What are two solutions to $\vec{X}' =A\vec{X}$? (What equations must you solve in order to find these solutions?)

10.

Suppose $\lambda = \beta i$ is an imaginary eigenvalue of A, with eigenvector $\vec{K}= i \vec{v}$. What are two solutions to $\vec{X}' =A\vec{X}$?

11.

Suppose $\lambda = \alpha + \beta i$ is a complex eigenvalue of A, with eigenvector $\vec{K}= \vec{u} + i \vec{v}$. What are two solutions to $\vec{X}' =A\vec{X}$?

12.

Suppose all solutions to the equation $\vec{X}' =A\vec{X}$ are periodic (oscillating with constant amplitude). What can you say about the eigenvalues/eigenvectors of A?

13.

Suppose all solutions of $\vec{X}' =A\vec{X}$ decay to the origin (i.e., both the x and y coordinates approach zero). What can you say about the eigenvalues of A?

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