A method of solving \(y^{(k)}-f(x)y=0\) (Q1186347)

A sufficient condition for \(\lim_{n\to\infty}(d^ k/dx^ k)(S_ n(x))=(d^ k/dx^ k)(\lim_{n\to\infty}S_ n(x))\), \(k\in\mathbb{N}\), to hold for a functional sequence \((S_ n(x))_{n\in\mathbb{N}}\), \(x\in[a,b]\), is given. This result is used to give a solution of the equation \(y^{(k)}(x)=f(x)y(x)\), \(x\in[a,b]\) in the form \(y(x)=\sum^ \infty_{n=1}f_ n(x)\), where \((f_ n(x))_{n\in\mathbb{N}}\), \(x\in[a,b]\) is defined by the formula \(f_ n^{(k)}(x)=f_{n-1}(x)\cdot f(x)\). Two examples for \(k=2\) are included.