Math 308 - Differential Equations Fall 2002
Homework Assignment 10
Due Friday, December 6
1. For each of the following second order differential equations, (i) use the method of undetermined
coefficients to find the general solution, and then (ii) solve the initial value problem.
(a) y00 + 5y0+ 4y= sin(2t), y(0) = 1, y0(0) = −2
(b) 3y00 + 8y0+ 5y=e−3tcos(t), y(0) = 0, y0(0) = 0
(c) y00 + 9y= 6 cos(3t), y(0) = 1, y0(0) = 3
2. For each of the following second order differential equations, determine if the method of undetermined
coefficients can be used to find yp. If so, write down the appropriate form for yp(t), but do not solve
for the coefficients.
(a) y00 + 5y0+ 4y=t2sin(3t)
(b) y00 + 8y0+y= ln(1 + t2)
(c) y00 + 2y0+y=tet
(d) y00 + 2y0+y=te−t
(e) y00 + 3y0+ 2y=1
1 + t2
(f) y00 + 9y= (t2+ 3) sin(2t)
(g) y00 + 4y0+ 4y= 8
3. (a) Suppose we have the equation
y00 +py0+qy =g1(t) + g2(t).(1)
Show that if yp1is a particular solution to
y00 +py0+qy =g1(t).
and yp2is a particular solution to
y00 +py0+qy =g2(t).
then yp1+yp2is a particular solution to (1).
(b) Use the result of (a) to solve the initial value problem
y00 + 3y0+ 2y= 2 sin(t) + 10 sin(2t), y(0) = −3, y0(0) = 0.
4. Show that the expression Acos(ωt)+Bsin(ωt) may be written as Rcos(ωt −φ), where R=√A2+B2,
and tanφ=B /A (assuming A6= 0). (The quantity Ris the amplitude and φis the phase of the
sinusoidal function.)
Hint: Use the trigonometric identity
cos(x−y) = cos xcos y+ sin xsin y
and choose φso that A=Rcos(φ) and B=Rsin(φ).
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