Exercises

1. Solve each of the following initial value problems.

   \(\frac{dy}{dt}=1.3y\), \(y(0)=-0.5\)
   \(\frac{dy}{dt}=-1.3y\), \(y(0)=0.5\)
   \(\frac{d^2y}{dt^2}=-1.3y\), \(y(0)=-0.5\), \(y'(0)=0\)
   \(\frac{d^2y}{dt^2}=-1.3y\), \(y(0)=0\), \(y'(0)=-0.5\)

2. Solve the initial value problem \(\frac{d^2y}{dt^2}=1.3y\), \(y(0)=-0.5\), \(y'(0)=0\). (Hint: Find two exponential functions, \(e^{rt}\) and \(e^{st}\), that satisfy the differential equation. Then look for a solution of the initial value problem of the form \(A\,e^{rt}+B\,e^{st}\).)
3. The frequency of household electric current is 60 hertz. (One hertz is one cycle per second.) If this current is described by a formula of the form \(x(t)=A\,\sin(\omega t)\), and time is measured in seconds, what is \(\omega\)?
4. The clock speed of a certain personal computer is 1.6 gigahertz. If the oscillation is described by a formula of the form

   \(x(t)=A\,\sin(\omega t)+B\,\cos(\omega t)\),

   and time is measured in microseconds, what is \(\omega\)? (The prefix giga- means one billion, and the prefix micro- means one-millionth.)
5. An airplane is flying directly overhead at an altitude of 20,000 feet. One minute later, the line of sight to the airplane makes an angle of 49 degrees with the ground. What is the approximate speed of the airplane in miles per hour?