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Cauchy-Euler equation with repeated roots One of several ways to find the second linearly independent solution of a Cauchy-Euler equation
a. What is the polynomial associated with this equation?
b. Show that if we let t = ex (or .x = ln t), then this equation becomes the constant coefficient equation
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.
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Chapter D2 Solutions
Calculus: Early Transcendentals, 2nd Edition
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