Solution of ordinary differential equations by series of delta functions (Q1892559)

Let \(y(x)\) be a distributional solution of the linear differential equation with polynomial coefficients \[ a_ n(x) y^{(n)}(x)+\cdots+ a_ 0(x) y(x)= 0\tag{1} \] and let \(u(x)= \sum^ \infty_{n= 0} a_ n \delta^{(n)}(x)\) be a formal solution of (1). Here \(\delta(x)\) is the Dirac delta function, \(\delta^{(n)}(x)\) is the distributional derivative of order \(n\) of \(\delta(x)\). The authors establish the relationship between \(u(x)\) and asymptotic expansion of \(y(\lambda x)\) as \(\lambda\to \infty\).