Someone solves as follows: "The auxiliary equation is

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Ordinary differential equations · Ordinary differential equations

Mathematical Methods for Physics, Ch Higher-order linear equations, the Wronskian and variation of parameters

Problem

Someone solves

y'' - 6y' + 9y = 0

as follows: "The auxiliary equation is

\lambda^2 - 6\lambda + 9 = (\lambda - 3)^2 = 0

, so

\lambda = 3

is a root, and also

\lambda = 3

is a root. Two roots give two solutions

y_1 = e^{3t}

and

y_2 = e^{3t}

, so the general solution is

y = c_1e^{3t} + c_2e^{3t}

, which can be written

y = Ce^{3t}

with

C = c_1 + c_2

an arbitrary constant." ((a)) Find the error in this working. ((b)) Compute

W(e^{3t}, e^{3t})

and use it to say what has gone wrong with the claimed pair of solutions. ((c)) Give the correct general solution, and verify by computing its Wronskian that the two functions in it are genuinely independent.

Related theory

Ordinary differential equations

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