Some Examples of the Second Order Equations in One Dimension
d/dx(adu/dx) = q for 0 < x < L
[Taken from J. N. Reddy's Book "Introduction to the Finite Element Method", J.N. Reddy, McGraw Hill Publishers, 2nd Edition, Page 71]
Field 
Primary Variable u 
Constant a 
Source term q 
Secondary Variable Qo 
Transverse Deflection of a Cable 
Transverse Deflection  Tension in Cable 
Heat Source Q 
Axial Force 
Axial Deformation of a bar 
Longitudiunal Dispalcement  EA (E= Young's Modulus, A = Cross Sectional Area) 
Distributed Transverse Load 
Axial Force 
Heat Transfer 
Temperature

Thermal Conductivity 
Friction or contact force on surface of bar 
Heat 
Flow Through Pipes 
Hydrostatic Pressure 
PD4/128m (D Diameter, m  viscosity) 
Flow Source (Generally Zero)  Flow Rate 
Laminar Incompressible Flow through a Channel under Constant Pressure Gradient  Velocity 
Viscosity  Pressure Gradient  Axial Stress 
Flow Through Porous Media 
Fluid Head 
Coefficient of Permeability  Fluid Flux  Flow (seepage) 
Electrostatics 
Electroststic Potential 
Dielectric Constant 
Charge Density 
Electric Flux 