Second-Order Linear Equations

1. Find the general solution of

   (a) $y'' - 2 y' + y = 6 {\rm e}^x$   Solution

   (b) $y'' - 4 y' + 4 y = x^{-3/2} {\rm e}^{2 x}$ for $x > 0$   Solution

   (c) $y'' + 4 y = \cos 2x$   Solution

   (d) $y'' + y' - 2 y = (x - 1) {\rm e}^x$   Solution

2. Solve the initial value problem $y'' + 6 y' + 9 y = 0$ with $y(0) = 6$, $y'(0) = -2$.   Solution

3. (a) Find a fundamental set of solutions for the differential equation $y'' - 2 y' + 2 y = 0$.   Solution

   (b) Find a particular solution of $y'' - 2 y' + 2 y = {\rm e}^{-x} \cos x$.   Solution

   (c) Solve the initial-value problem $y'' - 2 y' + 2 y = {\rm e}^{-x} \cos x$, $y(0)=0$, $y'(0)=1$.   Solution

4. Find the general solution of the differential equation $x^2 y'' + 2 x y' - 2 y = x^3$ for $x > 0$ given that $y = x$ is a solution of $x^2 y'' + 2 x y' - 2 y = 0$.   Solution
5. A mass of $m$ kilograms is attached to a spring with constant 1 newton/metre. In addition there is a dashpot which supplies a damping force whose magnitude in newtons is twice the speed of the mass. The mass is released from rest $0.1$ metres above its equilibrium position.

   (a) Find the position of the mass as a function of time if $m=2$.   Solution

   (b) For what value of $m$ will the amplitude of the steady-state response to an external force of $2 \cos t$ newtons be the greatest?   Solution

6. Consider a damped mass-spring system with mass $m = 1$ kg, damping coefficient $c = 2$ kg/s, and spring constant $k = 2 \mbox{kg}/\mbox{s}^2$.

   (a) If the mass is started at $t=0$ from the equilibrium position with a downwards velocity of 5 m/s, find the position for all subsequent $t$.   Solution

   (b) If an oscillating force of $5 \sin t$ newtons is applied to the system, find the steady state solution, and its amplitude.   Solution

7. An object of mass $0.5$ kg is attached to a spring with spring constant $k = 32$ newtons/metre and a damper with damping constant $c$, which is chosen so that the system is critically damped.

   (a) Find $c$.   Solution

   (b) The object is pushed upward $0.25$ metre and released with a velocity of 5 metres/sec downwards. When (if ever) will the object cross the equilibrium point?   Solution