Problem $1$

Consider a first$-$order system governed by the differential equation: $\backslash (\backslash$boldsymbol$\{$y$\}+\backslash$boldsymbol$\{\backslash$tau$\}\backslash$dot$\{\backslash$boldsymbol$\{$y$\}\}=\backslash$boldsymbol$\{$K$\}\backslash$boldsymbol$\{$x$\}(\backslash$boldsymbol$\{$t$\})\backslash ),$ where $\backslash (\backslash$boldsymbol$\{\backslash$tau$\}\backslash )$ and $\backslash (\backslash$boldsymbol$\{$K$\}\backslash )$ are time and sensitivity constants respectively, $\backslash (\backslash$boldsymbol$\{$x$\}\backslash )$ being the input and $\backslash (\backslash$boldsymbol$\{$y$\}\backslash )$ is the output.

a$)$ Determine the steady state error at the output of the system for a unit$-$ramp test$-$input.

b$)$ Determine the steady state output $\backslash (\backslash$boldsymbol$\{$y$\}\backslash )$ for the sinusoidal input Asin$\backslash (\backslash$omega $\backslash$boldsymbol$\{$t$\}\backslash ).$