University of Notre Dame
Aerospace and Mechanical Engineering

AME 301: Differential Equations, Vibrations and Control
Homework 1

B. Goodwine
J. Lucey
Fall, 2003
Issued: August 29, 2003
Due: September 3, 2003

For each of the following equations, identify them as

If the differential equation is a linear, homogeneous, constant coefficient ordinary differential equation, determine its solution (5 points).


  1. \begin{eqnarray*}
5 \ddot x + 6 \dot x + \sin(t) x &=& \cos(t^2)\\
x(0) &=& 1 \\
\dot x(0) &=& \pi.
\end{eqnarray*}

  2. \begin{eqnarray*}
2 \ddot x + 19 \dot x + 24 x &=& 0\\
x(0) &=& 1\\
\dot x(0) &=& 0
\end{eqnarray*}

  3. \begin{eqnarray*}
2 \frac{\partial^2 \zeta}{\partial \gamma^2} + 19 \frac{\parti...
...&=& \gamma^2 + \alpha^2\\
\zeta(0) &=& 1\\
\dot \zeta(0) &=& 0
\end{eqnarray*}

  4. \begin{eqnarray*}
6 \ddot x + 23 \dot x + x^3 &=& \sin(t^2)\\
x(0) &=& 1\\
\dot x(0) &=& 0
\end{eqnarray*}

  5. \begin{eqnarray*}
6 \ddot x + 23 \dot x + t^3 x^2 &=& 0\\
x(0) &=& 1\\
\dot x(0) &=& 0
\end{eqnarray*}

  6. \begin{eqnarray*}
6 \ddot x + 23 \dot x + t^3 x &=& \sin(t^2)\\
x(0) &=& 1\\
\dot x(0) &=& 0
\end{eqnarray*}

  7. \begin{eqnarray*}
2 \frac{d^2 \xi}{d \eta^2} + 19 \frac{d \xi}{d \eta} + 24 \xi &=& 0\\
\xi(0) &=& 1\\
\dot \xi(0) &=& 0
\end{eqnarray*}

  8. \begin{eqnarray*}
6 \ddot x + 23 \dot x + 3^2 x &=& \sin(t^2)\\
x(0) &=& 1\\
\dot x(0) &=& 0
\end{eqnarray*}

  9. \begin{eqnarray*}
\pi \ddot x + e \dot x + x &=& 0\\
x(0) &=& 1\\
\dot x(0) &=& 0
\end{eqnarray*}

  10. \begin{eqnarray*}
2 \frac{d^2 \zeta}{d \gamma^2} + 19 \frac{d \zeta}{d \gamma} + 24 \zeta &=& 0\\
\zeta(0) &=& 1\\
\dot \zeta(0) &=& 0
\end{eqnarray*}

  11. \begin{eqnarray*}
2 \frac{d^2 \zeta}{d \gamma^2} + 19 \frac{d \zeta}{d \gamma} +...
...4 \zeta &=& \sin \gamma\\
\zeta(0) &=& 1\\
\dot \zeta(0) &=& 0
\end{eqnarray*}


Last updated: August 29, 2003.
B. Goodwine (goodwine@controls.ame.nd.edu)