University of Notre Dame
Aerospace and Mechanical Engineering

AME 302: Differential Equations, Vibrations and Control
Homework 2

B. Goodwine
Spring, 2004		 Issued: January 30, 2004
Due: February 4, 2004

Unless otherwise indicated, all problems are from Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems, seventh edition.

  1. Consider
    \begin{displaymath} A = \left[ \begin{array}{cc} 1 & 3 6 & 1 \end{array}\right]. \end{displaymath}

    For this matrix, show that
    \begin{displaymath} A T = \left[ \lambda_1 \hat \xi_1 \quad \lambda_2 \hat \xi_2 \right]. \end{displaymath}

  2. Solve $\dot \xi = A \xi$ where
    \begin{displaymath} A = \left[ \begin{array}{cc} 1 & -3 6 & 1 \end{array}\right] \end{displaymath}

    and
    \begin{displaymath} \xi(0) = \left[ \begin{array}{c} 1 1 \end{array}\right] \end{displaymath}

    1. by using one eigenvalue/eigenvector pair and following the procedure in the book (with a, b, u and v); and,
    2. by computing both eigenvalue/eigenvector pairs and simply computing
      \begin{displaymath} \xi = e^{\lambda_1 t} \hat \xi_1 + e^{\lambda_2 t} \hat \xi_2. \end{displaymath}

    Are the answers the same?
  3. Section 7.7, number 1.
  4. Section 7.7, number 3.
  5. Section 7.7, number 5.
  6. Section 7.7, number 16.
  7. (Review) Consider
    \begin{displaymath} \dot x = 3 x \qquad x(0) = 1. \end{displaymath}

    Solve this equation by
    1. using the method from section 2.1 in the course text; and
    2. assuming a solution of the form $x(t) = e^{\lambda t}$, substituting and solving for $\lambda$.
  8. (Review) Consider
    \begin{displaymath} \dot x = -4 x + \sin(t), \qquad x(0) = 1. \end{displaymath}

    Solve this equation by
    1. using the method from section 2.2 of the course text; and,
    2. by using undetermined coefficients.
  9. (Review) Consider the system illustrated in the following figure, which is comprised of a board of uniform mass density per unit length of $\rho$. The board sits on top of two counter-rotating cylinders, each of which is rotating with an angular velocity of $\omega$. Other than rotating, the cylinders do not move. The coefficient of dynamic friction between the board and the cylinders is $\mu$.
    1. Determine the equation of motion for the system.
    2. If
      \begin{eqnarray*} x(0) &=& 1 \\ \dot x(0) &=& 0, \end{eqnarray*}

      determine $x(t)$.


2004-01-30