Y =  ( 

) 
and
A =  ( 

) 
Then we want to solve Y' = AY. The eigenvalues of A are 1±i, and the corresponding eigenvectors are
( 

) 
Therefore u(t) = e^{t}( A sin t + B cos t) and x(t) = e^{t}(A cos t + B sin t), where A, B are constants to be determined. However when t = 0, we have u(t) = x(t) = 10, so A = B = 10. Therefore u(t) = 10e^{t}(cos t  sin t) and x(t) = 10e^{t}(cos t + sin t). Finally the cat will hit the mirror when u(t) = 0, that is when t = π/4.