Is it possible to use the convolution theorem on a finite interval integral ? (Laplace)

Question

Say I have the following equation :

$$\int_{0}^{1}\cos(t-\tau)x(\tau) d\tau = t\cos(t)$$

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 of the professor ?

Answer 1

The equation has no solution.

Let $f(t) = \int_0^1 \cos(t-\tau ) x(\tau) d \tau $. Then we have $f'(t) = -\int_0^1 \sin(t-\tau ) x(\tau) d \tau $ 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 $g'' \neq g$, hence the equation has no solution.