Approximate Eigenvalues for Systems With Variable Parameters

The paper deals with an approximate analytical technique for solving the boundary-value problems governed by second-order differential equations with variable coefficients. The variable coefficient appearing in the system equation is expanded in ultraspherical polynomials in the desired interval to construct simple equivalent functions such that the approximate differential equations thus obtained have known closed form solutions. If the variable coefficient is approximated by a constant, the solutions are the sine and cosine functions, while a linear approximation leads to a solution in terms of Bessel functions of 1/3 order or Airy’s functions. In particular, the method has been applied to calculate the critical buckling load for a column of exponentially varying moment of inertia. Results obtained by present approach agree with exact results to a much better accuracy than other approximate analytical methods available in the literature. The technique is quite general and does not require any restriction on system parameters.