Past exam of the mathematics course of the University of Cambridge / 2013 / ia / Paper 2 / 7A (Differential Equations)

Retain the derivative marks and coefficients from the printed system. In matrix form it is $M\dot{u}+Ku=b(t)$, withSince $\det M=11$, multiplication by its inverse givesThe linear transformation $u=x+y$, $v=x-y$ diagonalizes this system:Both start at zero. An integrating factor gives $u=v=e^{-t}-e^{-3t}$, soDirect substitution into the two original equations confirms the different forcing coefficients; the exact cancellation in $y$ depends on those coefficients.