Solution of ordinary differential equations by series of delta functions
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 \(\lambda\to \infty\).
linear differential equation with polynomial coefficients
scientific article; zbMATH DE number 765148