Final Answer:
The roots of the indicial equation are -2 and -1/2. The Cauchy-Euler equation has the general solution y = x^(-2) (c1 + c2 x^3).
Explanation:
The first step in solving a Cauchy-Euler equation is to determine the roots of the indicial equation. The indicial equation for the given equation is 9x^2y' + 3xy + y = 0, which has the roots -2 and -1/2. The Cauchy-Euler equation is a linear second order differential equation with constant coefficients and can be solved using the method of undetermined coefficients. The general solution of the Cauchy-Euler equation is y = x^(-2) (c1 + c2 x^3), where c1 and c2 are arbitrary constants.
To find the particular solution of the given Cauchy-Euler equation, we need to use the method of variation of parameters. To find the particular solution, we have to solve the two first order linear differential equations formed by the two arbitrary constants c1 and c2. The auxiliary equation for this Cauchy-Euler equation is λ^2 – 9λ + 3 = 0. The roots of the auxiliary equation are λ1 = 3 and λ2 = -3.
To solve for the arbitrary constants c1 and c2, we use the two equations (3x^2y' + xy + y = 0) and (-3x^2y' + xy + y = 0). We solve the two equations for c1 and c2. The solution of the two equations is c1 = -1 and c2 = -2.
The particular solution of the Cauchy-Euler equation is therefore y = x^(-2) (c1 + c2 x^3) = x^(-2) (-1 + -2x^3). This is the solution of the Cauchy-Euler equation 9x^2y' + 3xy + y = 0.