Differential Equations and Linear Algebra, 4th Edition

A.1 Existence of Solutions

The approach we employ is the method of successive approximations, which was developed by the French mathematician Emile Picard (1856–1941). This method is based on the fact that the function y(x) satisfies the initial value problem in (1) on the open interval I containing $x=a$ if and only if it satisfies the integral equation

$$y(x)=b+{\displaystyle {\int }_a^{x}f(t,y(t))dt}$$ (4)

for all x in I. In particular, if y(x) satisfies Eq. (4), then clearly $y(a)=b,$ and differentiation of both sides in (4)—using the fundamental theorem of calculus—yields the differential equation $y^{\text{′}}(x)=f(x,y(x))$.

To attempt to solve Eq. (4), we begin with the initial function

$$y_0(x)\equiv b$$ (5)

and then define iteratively a sequence $y_1,y_2,y_3,\dots$ of functions that we hope will ...