Is it possible to use the convolution theorem on a finite interval integral ? int_0^1 cos(t-tau)x(tau)d tau=t cos(t)

duandaTed05 2022-10-20 Answered
I have the following equation :
0 1 cos ( t τ ) x ( τ ) d τ = t cos ( t )
if we replace 1 in the integral for t it is easily solvable using the convolution of Laplace and the answer will be
1 + 2 cos t
I'm studying for a test and have stumbled upon the equation above. Is there any way to solve the equation as it is written , or is it safe to assume its a mistake?
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Answers (1)

Messiah Trevino
Answered 2022-10-21 Author has 18 answers
The equation has no solution.
Let f ( t ) = 0 1 cos ( t τ ) x ( τ ) d τ. Then we have f ( t ) = 0 1 sin ( t τ ) x ( τ ) d τ and hence f ( t ) = f ( t ).
It follows that f ( t ) = a cos t + b sin t for some a,b and hence can not be equal to the right hand side above.
Alternatively: If we let g ( t ) = t cos t, we see that g g, hence the equation has no solution.
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