Internal problem ID [446]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Chapter 11 Power series methods. Section 11.2 Power series solutions. Page 624
Problem number: problem 33.
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 +2 \alpha y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 63
Order:=6; dsolve(diff(y(x),x$2)-2*x*diff(y(x),x)+2*alpha*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\alpha \,x^{2}+\frac {\alpha \left (\alpha -2\right ) x^{4}}{6}\right ) y \relax (0)+\left (x -\frac {\left (\alpha -1\right ) x^{3}}{3}+\frac {\left (\alpha ^{2}-4 \alpha +3\right ) x^{5}}{30}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 78
AsymptoticDSolveValue[y''[x]-2*x*y'[x]+2*\[Alpha]*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {\alpha ^2 x^5}{30}-\frac {2 \alpha x^5}{15}+\frac {x^5}{10}-\frac {\alpha x^3}{3}+\frac {x^3}{3}+x\right )+c_1 \left (\frac {\alpha ^2 x^4}{6}-\frac {\alpha x^4}{3}-\alpha x^2+1\right ) \]