Cauchy Euler Equations
A Cauchy – Euler equation is any equation of the formwhere andis a continuous function. The solution is written as the sum of two terms:
The solutionof the homogeneous equation(1) and a particular solutionof the non – homogeneous equationwhere the form ofdepends onand may be found using guesswork and intuition. If we have two boundary conditions then we can solve fro any constant to find the general solution.
To find the solutionassume a solution of the formSubstitute these into (1).
Simplify and factorise withto obtain
If we assumethenThis is called the indicial equation.
We can solve the above indicial equation into obtainandhence
Example: Fund the solution to(3) ifand when
The indicial equation is
We can solve this equation by factorising to obtain
for (3) so we do not need to look for a particular solution.
(4)+3*(5) givesthen from (4)
The solution is