Question $1(20$ pts$).$ The passive mechanical behavior of skeletal muscle can be modelled using

a simplified lumped parameter representation consisting of a linear spring $k_1$ in series with a

parallel linear spring$-$dashpot combination $k_2,b,$ as shown below:

One end of the muscle is fixed, and the displacement of the other end is $x.$ If $x_1$ denotes the

change in length from rest in spring $k1,$ then the forces in each element are given by:

$F_1=k_1{x}_1$

$F_2=k_2{x}_2$

$F_b=b\frac{d{x}_2}{dt}$

where $F_1,F_2$ and $F_b$ refer to elements $k_1,k_2$ and $b$ respectively, and $x_2=x-x_1.$

$($a$)$ If the length $x$ of the muscle is suddenly stepped and held from $0$ to $x_m$ at $t=0,$ solve the

system analytically for the applied force, $F.$

$($b$)$ If the applied force on the muscle $F$ is suddenly stepped and held from $0$ to $F_m$ at $t=0,$

find the analytical solution for the change in length, $x.$