An integral equation associated with linear homogeneous differential equations

Abstract

Associated with each linear homogeneous differential equation y(n)=∑i=0n−1ai(x)y(i) of order n on the real line, there is an equivalent integral equation f(x)=f(x0)+∫x0xh(u)du+∫x0x[∫x0uGn−1(u,v)a0(v)f(v)dv]du which is satisfied by each solution f(x) of the differential equation.

Keywords

• linear homogeneous differential equations
• integral equations
• initial value problems
• variation of parameters formula
• uniform convergence.