Final answer:
The student's question involves proving a differential equation, which resembles the Cauchy-Euler equation, through methods such as substitution and simplification.
Step-by-step explanation:
The student's question is about proving a given differential equation, specifically x²y''(x)-2xy'(x)+2yx=0. To prove this, we can check if it is a characteristic of a certain type of differential equation. Upon inspection, this equation resembles the format of the Cauchy-Euler equation, which is of the form a_nx^n y^{(n)} + a_{n-1}x^{n-1} y^{(n-1)} + … + a_1xy' + a_0y = 0, where the coefficients a_n are constants, and y^{(n)} denotes the nth derivative of y.
If we assume a solution of the form y = x^m, then y' = mx^{m-1} and y'' = m(m-1)x^{m-2}. Substituting these into the original differential equation, we can determine the value of m that satisfies the equation. This is a standard method for solving Cauchy-Euler equations and can be done through substitution and simplifying by equating coefficients from both sides of the equation