Respuesta :
1/2 ∫₀t f( t - T ) sin ( 2T ) dT is the given differential equation with initial conditions x(0).
What does Laplace transform mean?
The Laplace transform transforms a given derivative function with a real variable t into a complex function with a variable s.
Let f(t) be given for t 0 and presume that it complies with a number of later-explained requirements.
Take the Laplace transform , we get
s² X(s) - sx(0) - x'(0) + 4X (s ) = F(s)
(s² + 4 ) X(s ) = F(s)
X(s) = F(s)/s² + 4
x(t) = L⁻¹ { F(s)/s² + 4 }
= 1/2 L⁻¹ { F(s) 2/s² + 4 }
= 1/2 [ f(t) * sin (2t) ]
= 1/2 ∫₀t f( t - T ) sin ( 2T ) dT
Learn more about Laplace transform
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