 
Summary: EXAMPLE OF OSCILLATIONS FORCED AT NATURAL
FREQUENCY
MATH2574H, SPRING 2012, UNIVERSITY OF MINNESOTA
GREG W. ANDERSON
Problem
Solve the initial value problem
y + 9y = 24 cos(3t), y(0) = 2, y (0) = 9.
Calculations
Step 1: General solution of homogeneous equation.
yh = C1 cos(3t) + C2 sin(3t).
Step 2: Particular solution of given equation (ignoring initial conditions).
0.0.1. First (bad) guess.
yp = A cos(3t) + B sin(3t)
might be reasonable but fails because it solves the homogeneous equation.
0.0.2. Second (good) guess.
yp = At cos(3t) + Bt sin(3t).
(In general multiplying by t is recommended to get out of trouble. That's not the
whole story but often it works.) To quickly get second derivative let's remember
the iterated Leibniz rule: (fg) = f g + 2f g + fg . Okay, let's go:
yp = 6A sin(3t)  9At cos(3t) + 6B cos(3t)  9Bt sin(3t).
