Biharmonic Equation

Author

Eric W. Weisstein
November 17, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialEquations/BiharmonicEquation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/BiharmonicEquation.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Solution

Homogeneous

Expand[Together[Nest[(1/r)D[r D[#,r],r]&,ϕ[r],2]]]

DSolve[((ϕ'[r])/(r^3))-((ϕ''[r])/(r^2))+((2 Derivative[3][ϕ][r])/r)+Derivative[4][ϕ][r]==0,ϕ[r],r]//Simplify

Simplify[((ϕ'[r])/(r^3))-((ϕ''[r])/(r^2))+((2 Derivative[3][ϕ][r])/r)+Derivative[4][ϕ][r]/.ϕ:>Function[r,(1/2) (r^2) C[2]-(1/4) (r^2) C[3]+C[4]+C[1] Log[r]+(1/2) (r^2) C[3] Log[r]]]

Inhomogeneous

DSolve[((ϕ'[r])/(r^3))-((ϕ''[r])/(r^2))+((2 Derivative[3][ϕ][r])/r)+Derivative[4][ϕ][r]==64β,ϕ[r],r]//Simplify

Simplify[((ϕ'[r])/(r^3))-((ϕ''[r])/(r^2))+((2 Derivative[3][ϕ][r])/r)+Derivative[4][ϕ][r]/.ϕ:>Function[r,(r^4) β+(1/4) (r^2) (2 C[2]-C[3])+C[4]+(C[1]+(1/2) (r^2) C[3]) Log[r]]]