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Differential Equations –Important Applications
First Order
Unforced:
τ
dx/dt + x =
0
Solution:
x = x(0)*exp( t/
τ
)
where x(0)= initial condition at t=0
Forced:
τ
dx/dt + x = x
in
x
in
= constant input forcing
Solution:
x=x(0) *exp(t/
τ
) + x
in
*(1exp(t/
τ
))
where x(0)=initial condition at t=0, if
x(0)=0
then the first term vanishes.
Numerical solution form:
dx/dt = (x
in
x)/
τ
x(0)=initial condition
Second Order
Unforced:
d
2
x/dt
2
+ 2*
ζϖ
n
*dx/dt+
ϖ
n
2
*x = 0
for dx/dt(0)=0
Solution:
x=x(0)*exp(
ζϖ
n
*t)*(cos(
ϖ
r
*t)+
ζ
*sin(
ϖ
r
*t)/sqrt(1
ζ
2
))
where
ϖ
r
=
ϖ
n*sqrt(1
ζ
2
)
Forced:
d
2
x/dt
2
+ 2*
ζϖ
n
*dx/dt+
ϖ
n
2
*x = x
in
*
ϖ
n
2
x
in
=constant input forcing
Solution:
x=unforced +
x
in
*(1 exp(
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 '07
 Yamashiro

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