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)

I have the following equation :
${\int }_{0}^{1}\mathrm{cos}\left(t-\tau \right)x\left(\tau \right)d\tau =t\mathrm{cos}\left(t\right)$
if we replace 1 in the integral for t it is easily solvable using the convolution of Laplace and the answer will be
$-1+2\mathrm{cos}t$
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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Messiah Trevino
The equation has no solution.
Let $f\left(t\right)={\int }_{0}^{1}\mathrm{cos}\left(t-\tau \right)x\left(\tau \right)d\tau$. Then we have ${f}^{\prime }\left(t\right)=-{\int }_{0}^{1}\mathrm{sin}\left(t-\tau \right)x\left(\tau \right)d\tau$ and hence ${f}^{″}\left(t\right)=-f\left(t\right)$.
It follows that $f\left(t\right)=a\mathrm{cos}t+b\mathrm{sin}t$ for some a,b and hence can not be equal to the right hand side above.
Alternatively: If we let $g\left(t\right)=t\mathrm{cos}t$, we see that ${g}^{″}\ne g$, hence the equation has no solution.