Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 2 / 17 / d

Take the switch-off time to be $t=0$. The state remains continuous through the source step, so $\mathbf{y}(0)=(1,1,1/4{)}^T$. Its eigendirection decomposition isThe antisymmetric mode has zero amplitude because the first two initial components are equal. Applying the decay factors from part (b),The first two responses decay at rate one and the third at rate four, approaching the new zero-source steady state.