What does "\$\cos(2t)u(t)\$" stand for in this context?
You're given the input current when a 1V step is  applied, hence you can find the differential equation defining the LTI system. From this you can determine the response to cos(2t)u(t), either by solving the differential equation or via the Laplace transform.

Alternatively, since initial conditions are clearly zero, determine the TF: \$\frac{I(s)}{V(s)}\$ by Laplace transforming \$\small i(t)\$, and noting that the Laplace transform of the input is \$\small V(s)=\large \frac{1}{s}\$. 

\$ s \rightarrow j\omega\$ will then allow you to determine the steady state response to the given sinusoid.