this case will lead to similar problem that weve

Question

This case will lead to similar problem that we've had all other time we've run in double roots or double eigenvalues. We only find a single solution and will require a second solution. In this case this can be demonstrated as the second solution will be,

y₂(x) = x^r In x

and therefore the general solution for this case is,

 y(x) = c₁ x^r + c₂ x^r ln x = x^r (c₁ + c₂ ln x)

 We can again see a purpose for requiring x>0. If we didn't we'd contain all sorts of problems along with this logarithm.