Internal problem ID [726]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Chapter 5.2, Series Solutions Near an Ordinary Point, Part I. page 263
Problem number: 21.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-2 y^{\prime } x +\lambda y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 63
Order:=6; dsolve(diff(y(x),x$2)-2*x*diff(y(x),x)+lambda*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {\lambda \,x^{2}}{2}+\frac {\lambda \left (\lambda -4\right ) x^{4}}{24}\right ) y \relax (0)+\left (x -\frac {\left (\lambda -2\right ) x^{3}}{6}+\frac {\left (\lambda -2\right ) \left (-6+\lambda \right ) x^{5}}{120}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 80
AsymptoticDSolveValue[y''[x]-2*x*y'[x]+\[Lambda]*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {\lambda ^2 x^5}{120}-\frac {\lambda x^5}{15}+\frac {x^5}{10}-\frac {\lambda x^3}{6}+\frac {x^3}{3}+x\right )+c_1 \left (\frac {\lambda ^2 x^4}{24}-\frac {\lambda x^4}{6}-\frac {\lambda x^2}{2}+1\right ) \]