Exact analytic method for solving variable coefficient differential equation (Q1177985)

The solution of differential equation with variable coefficients \[ Aw(x)=\sum^ k_{m=0\atop {0\leq v_ m+u_ m\leq k}}(P_ m(x)w(x)^{(v_ m)})^{(u_ m)}=f(x)\leqno(1) \] is considered for arbitrary boundary conditions, \(u_ m\), \(v_ m\) being positive integers. The following analytical method is suggested. Let the operator \(A^{-1}\), inverse to \(A\), exist for arbitrary function \(f\in L_ 2[x_ 0,x_ N]\) and given boundary conditions. Let us divide the interval \([x_ 0,x_ N]\) into \(N\) elements \(\Delta_ i=[x_{i- 1},x_ i)\). In \(\Delta_ i\) the equation (1) can be converted into one with constant coefficients: \[ \tilde A\tilde w(x)=\sum^ k_{m=0\atop {0\leq u_ m+v_ m\leq k}}(P_ m(\tilde x_ i)\tilde w(x)^{(v_ m)})^{(u_ m)}=f(x)\leqno(2) \] where \(\tilde x_ i=(x_{i-1}+x_ i)/2\), \(P_ m(x)\in\mathbb{C}^{\hbox{max}(u_ m,v_ m)}(\Delta_ i)\). It is shown that the solution of (2) converges to that of (1). Four numerical examples with elementary solutions are given for demonstrating that the suggested analytical method gives satisfactory results.