How Do You Solve an Euler-Cauchy Equation Like x^2y'' + xy' - y = 1/x?

[engineer_dave]

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

 

[HallsofIvy]

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

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.