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NonHomogeneous Equations
y(x) = y
c
+ y
p
if f(x)
≠
y
c
y
p
= A*f(x) (when f(x) is exp or
polynomial)
y
p
= Asin(x) + Bcos(x) (when f(x) is
sin/cos)
polynomial terms: (Ax
2
+ Bx + C)
even if some of the coefficients are
0
if f(x) = y
c
multiply “bad” term by smallest
power of x that will eliminate
duplication
Variation of Parameters
If y’’ + P(x)y’ + Q(x)y = f(x) &
y
c
=c
1
y
1
+ c
2
y
2
Y
p
= y
1
∫
[y
2
f(x))/w] + y
2
∫
[y
1
f(x)]/w
Where w=(y
1
)(y’
2
) – (y
2
)(y’
1
)
Undamped Forced Oscillation
mx’’ + kx = F
0
cos(
ϖ
t);
ϖ
0
=
√
(k/m)
x
c
= c
1
cos(
ϖ
0
t) + c
2
sin(
ϖ
0
t)
x
p
= [(F
0
/m)/(
ϖ
0
2

ϖ
0
)]cos(
ϖ
t)
if
ϖ
0
=
ϖ
resonance
ϖ
0
is internal &
ϖ
is external
frequency
Damped Forced Oscillations
mx’’ + cx’ + kx = F(t)
x
c
is the transient solution
x
p
= stable solution
amplitude of x
p
is called forced
amplitude. It’s max value is called
practical resonance
Method of Elimination
Solve one equation in terms of
another
Take the derivative of one equation
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 Spring '07
 Stenstones?
 Equations

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