The distribution solutions of ordinary differential equation with polynomial coefficients (Q5943070)

The author studies the type of solutions of differential equations of the form NEWLINE\[NEWLINEty^{(n)}(t)+ my^{(n-1)}(t)+ ty(t)= 0,NEWLINE\]NEWLINE where \(m\) is any integer and \(n\geq 2\) for \(t\in(-\infty, \infty)\). He obtains the following cases:NEWLINENEWLINENEWLINE(1) If \(m>n\), then all solutions of such an equation are singular distributions of the \(\delta\) distribution with its derivative, or the solutions contain both singular distributions and classical ones.NEWLINENEWLINENEWLINE(2) If \(m=n\), then the solution of such an equation is only the singular distribution of \(c\delta\) where \(c\) is any constant.NEWLINENEWLINENEWLINE(3) If \(m<n\), then all solutions of such an equation are classical, that are continuous functions.