Cauchy-Euler equation

In mathematics, a Cauchy-Euler equation (also Euler-Cauchy equation) is a second order ordinary differential equation of the form

Related Topics:
Mathematics - Ordinary differential equation

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:x^2u+bxu'+cu=0.,

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

These differential equations have one relatively simple solution xα. Observe

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:x^2(x^lpha)+bx(x^lpha)'+cx^lpha=x^2(lpha(lpha-1)x^{lpha-2})+bx(lpha x^{lpha-1})+cx^{lpha},

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:=lpha(lpha-1)x^lpha+blpha x^{lpha}+cx^{lpha},

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:=left(lpha(lpha-1)+blpha+c ight)x^lpha,

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:=left(lpha^2+(b-1)lpha+c ight)x^lpha.,

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Since xα is zero only when x is zero for positive α (which corresponds to a trivial solution) and never zero for negative α, we consider where the quadratic in α is zero. This has roots where

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:lpha={1over 2}left(-b+1pmsqrt{b^2-2b-4c+1} ight).

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

However, if the two roots are equal, we need to introduce a factor to obtain two linearly independent solutions. Observe if we take the product of ln x and xα,

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:x^2(x^lphaln{x})+bx(x^lphaln{x})'+cx^lphaln{x}.,

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

On expanding, we get

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:x^lphaleft((ln{x})(lpha^2 + (b - 1)lpha + c) + 2lpha+ b - 1 ight).

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Now if the two roots are equal, this means b2-2b-4c+1 = 0. So, substitute α = 1/2(1-b),

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:x^{{1over 2}(1-b)}left(2cdot{1over 2}(1-b)+b-1 ight)=0.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

So xα ln x is a solution where the roots of the quadratic above are equal.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~