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Find the general solution for the following Cauchy-Euler equation:

(a) [tex] x^5 y^{(5)} + 12x^4 y^{(4)} + 17x^3 y^{(3)} - 39x^2 y'' + 103x y' - 128y = 0 [/tex]

Answer :

Final answer:

The general solution for the Cauchy-Euler equation can be found by employing variable substitution to transform it into a standard linear differential equation.

Explanation:

The equation you've provided is an example of a Cauchy-Euler equation, which is a type of homogenous differential equation. In these equations, each term is a derivative of the function and contains a factor of x that is exactly one less than its order of derivative. In order to find the general solutions to these types of equations, we use the method of variable substitution, specifically, by setting x = ez. The equation then transforms from a Cauchy-Euler equation to a standard linear differential equation, which is easier to solve and provides a more general solution. After obtaining a solution this way, we then replace the variable z with ln|x| to revert back to the original variables of the problem.

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