Ch 09/Cauchy-Euler Equations/H.K. Dass 3.28

Cauchy-Euler Equations

A Cauchy-Euler equation has variable coefficients as powers of x. Substitute x = e^z to convert it into a constant-coefficient equation.

IThe Transformation

Form: .


Put (so , ).


Rules: , , .


After substitution, solve the constant-coefficient equation in , then replace .

Example 1easy

Solve .

Show step-by-step solution+
Substitute
. C.F. .
P.I.
.
Answer
Example 2medium

Solve .

Show step-by-step solution+
Substitute
. Roots: .
P.I. fails (m=4 is root)
Multiply by : P.I. .
Answer
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