The discussion focuses on solving the Euler-Cauchy equation represented by x²y'' + xy' - y = 1/x. The correct approach involves identifying the coefficients a and b, where a should be +1 instead of -1. After determining the roots using the characteristic equation m(m-1) + am + b = 0, users can apply the method of variation of parameters for the particular solution. Alternatively, the change of variable u = ln(x) can simplify the equation to a constant coefficients form.
PREREQUISITESStudents studying differential equations, mathematicians solving advanced calculus problems, and educators teaching methods for solving Euler-Cauchy equations.
One obvious error: a= +1 here , not -1. Yes, you can use "variation of parameters". Also, because the right hand side is a power of x, you could use "undertermined coefficients" although it is slightly harder to "guess" the correct form for a particular solution in such an equation as compared to a "constant coefficients" equation. Finally, the change of variable u= ln(x) will convert this equation to a "constant coefficients" equation.engineer_dave said:Homework Statement
Find the general solution of x^2y" + xy' - y = 1/x
Homework Equations
m(m-1) +am + b = 0 to solve an Euler Cauchy equation
The Attempt at a Solution
a=1 b=-1
m(m-1) -m -1 =0
m^2 - 2m -1 = 0
I just want to know whether the first step is right. And once I find out the values of M, do I use variation of parameters to find the particular solution? Thanks