## A method for the approximate solution of the second-order linear differential equations in terms of Taylor polynomials.(English)Zbl 0887.65084

Given a second-order linear differential equation for $$y:x \mapsto y(x)$$ $P(x)y'' +Q(x)y' +R(x)y= f(x) \tag{1}$ where $$P,Q,R,f$$ are elements of $$C^n([a,b], \mathbb{R})$$ with $$P(x)\neq 0$$ $$\forall x\in[a,b]$$ and associated conditions of the form $\sum^1_{i=0} \bigl[a_i y^{(i)} (a)+b_iy^{(i)} (b)+c_iy^{(i)} (c)\bigr]= \lambda \tag{2}$
$\sum^1_{i=0} \bigl[\alpha_iy^{(i)} (a)+ \beta_iy^{(i)} (b)+ \gamma_i y^{(i)} (c) \bigr]= \mu$ where $$a\leq c\leq b$$ and $$a_i,b_i,c_i, \alpha_i, \beta_i, \gamma_i, \lambda$$ and $$\mu$$ are suitable real coefficients.
The author is interested in an approximate solution of problem (1), (2) in terms of Taylor polynomials about any point $y(x)= \sum^N_{n=0} {1\over n!} y^{(n)} (c)(x-c)^n$ with $$a\leq c\leq b$$ and $$y^{(n)} (c)$$, $$n=0,1,\dots,N$$ are the coefficients to be determined. A matrix method is introduced to determine the Taylor polynomial of order $$N$$ of a solution to (1), (2). The method is illustrated by considering four examples.