The bilinear equation with the essential boundary condition of u(0) = U0 and natural boundary condition of at x = L , and the domain 0 < x <L appears in several forms in engineering.
CABLE(ROPE) UNDER A TRANSVERSE LOAD u = deflection a= tension q(x)= transverse load (eg: weight of cable) Q = Axial force BAR IN TENSION u = deflection a= EA q(x)= Friction(Traction on the bar) Q = Axial force HEAT TRANSFER u = Temperature a= Thermal conductivity(k) q(x)= Heat generated Q = Heat flux ONE DIMENSIONAL LAMINAR INCOMPRESSIBLE FLOW (Gradient(p)=const) u = velocity a= viscosity (η) q(x)= Pressure gradient Q = Axial stress FLOW IN POROUS MEDIA IN ONE DIMENSION u = Fluid head a= Permeability const q(x)= flux Q = Flow ELECTROSTATICS u = Electrostatic potential a= dielectric constant q(x)= charge density Q = Electric Flux 
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