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lilmoore11p8 2022-07-15 Answered
Consider the differential equation
x ˙ ( t ) = α x ( t ) + β u ( t ) ,
with α , β > 0 positive constant scalars and initial condition x ( 0 ) = x 0 . Function u ( ) is known. The Laplace solution for this equation should be
x ( t ) = e α ( t t 0 ) x 0 + β 0 t e α ( t τ ) u ( τ ) d τ .
However, if now I take the time derivative, I get
x ˙ ( t ) = α e α ( t t 0 ) x 0 + β u ( t ) .
Comparing to the first equation, this leaves me with
x ( t ) = e α ( t t 0 ) x 0 ,
which looks inconsistent with the Laplace solution. Anybody has a clue on what's going on here?
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Answers (1)

Kayley Jackson
Answered 2022-07-16 Author has 16 answers
The mistake is that t is also inside the integral.
To get things right, write
d d t 0 t e α ( t τ ) u ( τ ) d τ = d d t [ e α t 0 t e α τ u ( τ ) d τ ]
and use the product rule.
This gives
= α e α t 0 t e α τ u ( τ ) d τ + e α t e α t u ( t ) = α 0 t e α ( t τ ) u ( τ ) d τ + u ( t )
which allows you to check that you have a solution of the ODE.
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