Math 277 01,

the course of Dr. Mihailovs

Midterm 3

November 20, 1998

  1. Determine values of $ b$, for which the oscillator $ \Ddot{x}+2b\Dot{x}+x=0$ is overdamped, underdamped, or critically damped.

  2. For what value of $ \omega$ the harmonic oscillator $ \Ddot{x}+x=\cos\omega t$ has resonance solutions?

  3. Determine whether the system $ \Dot{x}=2x^2\cos y+6y\cos x$, $ \Dot{y}=3y^2\sin x-4x\sin y$ is Hamiltonian. If so, find a Hamiltonian function.

  4. A $ 2\times 2$ matrix $ A$ has an eigenvalue 3. Find another eigenvalue of $ A$ if the system $ \Dot{\bf x}=A{\bf x}$ is Hamiltonian.

  5. Solve the initial-value problem $ y'=\frac{2x^2+y^2}{xy}$, $ y(1)=1$ .

  6. Solve the initial-value problem $ 2xy'+y=x^2y^5$, $ y(1)=1$ .

  7. Solve the difference initial-value problem $ a_{n+1}=a_n+6a_{n-1}$, $ a_0=2$, $ a_1=1$ .

  8. Solve the difference equation $ a_{n+1}=a_n+6a_{n-1}-6$.

  9. Solve the initial-value difference problem $ a_{n+1}=5a_n+4b_n$, $ b_{n+1}=b_n-a_n$, $ a_0=2$, $ b_0=3$ .

Copyright © 1998 Aleksandrs Mihailovs