Solution of ordinary constant coefficient linear differential equations

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Solution of ordinary constant coefficient linear
differential equations

• Second Order Critically Damped and second order
  underdamped cases

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Transforming
Given the following differential equation,
determine P.
Where
Transforming
The characteristic equation is
Which has repeated roots and is solved as follows
Multiplying through by s and letting s0
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Solving for fractions
Multiplying through by
and letting
Special procedures are required to solve for c
for repeated roots
Multiply through by
and solve for each order of s
For
For
For
Solving for C
Taking the inverse transform, then
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Response Plot
Plotting, we get a non-oscillatory exponential
response
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• Second Order Under-damped Case

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Transforming
Given the following differential equation,
determine P.
Where
Transforming
The characteristic equation is
Which has imaginary roots and is solved as
follows
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Solving for fractions
Multiplying through by s and letting s0
Multiplying through by
And letting
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Inverting
Performing the inverse transform (see pages 29,30
and 31 of SC)
for
Where
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Response plot
Plotting, we get an oscillatory exponential
response
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Matlab solution
num1 den1 1 1 step(num,den)