Using a Laplace transform to figure something out in a DC circuit
The LaPlace transform isn't really used in DC analysis of circuits.  Typically, you would just use your standard "Circuits 101" tools to analyze the DC properties of a circuit.  These are:


Kirchhoff's Current Law (Nodal Analysis)
Kirchhoff's Voltage Law (Loop Analysis)
Thevenin's (or Norton's) Theorem


In DC, capacitors are treated as open circuits (hence why they are also known "DC blockers") and inductors are treated as close circuits (or dead shorts).

LaPlace transforms are used in both frequency and transient analysis.  They replace the more complex method of solving differential equations, or using convolution to determine the response of a circuit to an input.

Transient analysis builds upon the basic DC analysis, but determines what the circuit will do in response to a transient (a step input, an impulse, a ramp, etc).  

Frequency analysis then builds upon the transient analysis to determine the response of a circuit to periodic signals (sine wave, square wave, triangle waves, as an example).

LaPlace analysis is frequently used in 


Filter design (creating a circuit with specific frequency responses)
Controls (forcing a system or circuit to react with particular transient and frequency responses).
Higher level analysis of most electrical engineer (power, communications, electromagnetics,etc).