Abstract
A simple boundary value problem for a n-th order linear ordinary differential
equation which appears typically in the theory of Heaviside cable and Thomson
cable is treated. Output-input voltage relation is investigated. We found the best constant
of Sobolev-type inequality, which estimates the square of supremum of absolute
value of output voltage from above by the power of input voltage. The best constant
is a rational function of the characteristic roots and also a rational function of the
characteristic coefficients. The second formula for the best constant is very important
because even for small number of n it is difficult to obtain the exact value of characteristic
roots but in some cases it is easy to know the coefficients of characteristic
polynomial. Giambelli’s formula which appears in the theory of representation of finite
groups plays an important role.
