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The surface integral T is taken over the area G of the plate. , the local elastic potential integrated over the domain G. 10) 2(1 ; ) uxx + uyy + uxy ; uxxuyy : The total elastic energy is then given by integration over the volume of the plate. Integration rst over z 2 dz alone from ;h(x y)=2 to +h(x y)=2 delivers h3 (x y)=12. 12) This term describes the work done by the external forces when the points on the plate are displaced by the displacement u. 16) This is the plate equation for weakly varying thickness h(x y).

Near a singular point the Frobenius method will be used. 32). Now b(x ; x0 )=a(x ; x0 )2 will be replaced by p1 (x). Thus if a(x ; x0 ) = 0, then p1 (x) = 1 (singular point pole). A function regular everywhere but with one pole at x0 can no longer be expanded into a power series, but it can be represented by a Laurent P n series 1 n=;1 an (x ; x0 ) . ) If an = 0 for n < ;m < 0 a;m 6= 0, one says that the point x0 is a pole (a regular singular point) of order m. Singular points that are not poles are called irregular singular or essential singular.

The series is convergent. Since the method of power series is just the special case = 0 of the Frobenius method, we will discuss only the latter. 32) using the Frobenius method. 37) n=0 since a convergent power series may be di erentiated. 38) A power series vanishes only if all coe cients vanish. 39) since a0 6= 0 cancels. The case n 6= 0 will be treated later. 39) is the so-called indicial equation. 33). 42) The command DSolve[xˆ2*y00[x]+3*x*y0 [x]-3*y[x]==0, y[x],x] delivers the same result.