Time-domain to s-domain conversion: Convert the following time-domain mathematical representations (i.e. ordinary differential equations) of real-world systems into the s-domain. Please keep in mind that the initial conditions are not all zero.

a. mx + kx = 0, x(0) = 1 m, x(0) = 0.1 m/s.
b. mx + cx + kx = F(t), x(0) = 1 m, x(0) = 0 m/s Here F(t) represents an external force acting on the mass-spring-damper system. Such external forces (or other signals) are referred to as inputs.
c. RCv + v = v(t), v(0) = 1.5 V.
This represents a mathematical model for an electrical system. The actual system (electrical), the output variable (voltage v(t)), or the input (supply voltage vo(t)) should not affect your solution procedure. Indeed, later we will see that other systems (such as thermal systems) can be modeled with a very similar looking differential equation.
d. mx +cx+kx = củ + ku, x(0) = 1 m, x(0) = 0 m/s, u(t) = 0.05 sin(t).
This is similar to the car suspension example discussed in class, except that now I have told you that the road "waviness" can be modeled as a sine wave. Again u(t) represents an input to the car suspension system, and the vertical motion x(t) of the car represents an output.