Euler's homogeneous differential equation - revisited (Q1061285)

The author obtains an explicit formula in terms of an n-fold integral for a solution of the generalized Euler equation \(\prod^{n}_{j=1}(xd/dx+p_ j)y(x)=f(x)\) where \(p_ j\), \(j=1,2,...,n\) are real constants and \(x^{r-1}f(x)\to L\) as \(x\to 0+\), with \(r=\min\) \(\{p_ 1,...,p_ n\}\). This solution is defined for \(x>0\) and satisfies \(x^{s-1}y(x)\to L\delta_{rs}\) where \(s=\max \{p_ 1,...,p_ n\}\), \(\delta_{rs}=0\) if \(r\neq s\), \(\delta_{rs}=1\) if \(r=s\). The basic idea is to treat the case where \(n=1\) and then treat the nth order equation above as the system \((xd/dx+p_ j)z_ j(x)=z_{j+1}(x),\) \(j=1,2,...,n\) with \(z_{n+1}(x)=f(x)\).