Chapter D2.2, Problem 62E

Cauchy-Euler equation with repeated roots One of several ways to find the second linearly independent solution of a Cauchy-Euler equation $t^2{y}^{″}\left(t\right)+at{y}^{\prime }\left(t\right)+by\left(t\right)=0$, for t > 0 in the case of a repeated real root is to change variables.

a.    What is the polynomial associated with this equation?

b.    Show that if we let t = e^x (or .x = ln t), then this equation becomes the constant coefficient equation

$y^{″}\left(x\right)+\left(a-1\right)y^{\prime }\left(x\right)+by\left(x\right)=0.$

c.    What is the characteristic polynomial for the equation in part (b)? Conclude that if the polynomial in part (a) has a repeated root, then the characteristic polynomial also has a repeated root.

d.    Write the general solution of the equation in part (b) in the case of a repeated root.

e.    Express the solution in part (d) in terms of the original variable t to show that the second linearly independent solution of the Cauchy-Euler equation is y = t^(1–a)/2 ln t.