Math 333 Logarithms Worksheet With Answers
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13Math 333
Problem Set #7
– Solution
18 April 2003
A1. (a)
Solve the initial value problem
y + 2y + 10y = 37 cos 3t;
y(0) = 0,
y (0) = 1.
Identify which terms in your solution belong to the transient solution and which
belong to the steady-state solution.
2
Solution: The characteristic equation is r
+ 2r + 10 = 0; the roots are
1
3i,
so the complementary solution has the form
y
= e (c
cos 3t + c
sin 3t).
1
2
The particular solution has the form
Y
= A cos 3t + B sin 3t.
From this we get
Y
=
3A sin 3t + 3B cos 3t
Y
=
9A cos 3t
9B cos 3t.
Making the substitutions in the original DE gives
( 9A + 6B + 10A) cos 3t + ( 9B
6A + 10B) sin 3t = 37 cos 3t.
This gives us the system
A + 6B = 37
6A +
B =
0
The solution is A = 1, B = 6.
The solution to the initial value problem now reads
y = cos 3t + 6 sin 3t + e (c
cos 3t + c
sin 3t).
1
2
from this we get
y
=
3 sin 3t + 18 cos 3t
e (c
cos 3t + c
sin 3t) +
1
2
e ( 3c
sin 3t + 3c
cos 3t).
1
2
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