The bi-linear 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.

u = deflection

a= tension

q(x)= transverse load (eg: weight of cable)

Q = Axial force

u = deflection

a= EA

q(x)= Friction(Traction on the bar)

Q = Axial force

u = Temperature

a= Thermal conductivity(k)

q(x)= Heat generated

Q = Heat flux

u = velocity

a= viscosity (η)

q(x)= Pressure gradient

Q = Axial stress

u = Fluid head

a= Permeability const

q(x)= flux

Q = Flow

u = Electrostatic potential

a= dielectric constant

q(x)= charge density

Q = Electric Flux

**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