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Ordinary differential equations

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

These differential equations have one relatively simple solution x^α. Observe

Since x^α is only zero 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

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 x^α,

On expanding, we get

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

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

Example

Given

we substitute the simple solution x^α:

For this to indeed be a solution, either x=0 giving the trivial solution, or the coefficient of x^α is zero, so solving that quadratic, we get α=1,3. So, the general solution is

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