Internal problem ID [4499]
Book: Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson
2018.
Section: Chapter 8, Series solutions of differential equations. Section 8.3. page 443
Problem number: 6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x^{2}-1\right ) y^{\prime \prime }+\left (1-x \right ) y^{\prime }+\left (x^{2}-2 x +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 54
Order:=6; dsolve((x^2-1)*diff(y(x),x$2)+(1-x)*diff(y(x),x)+(x^2-2*x+1)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{12} x^{4}-\frac {1}{15} x^{5}\right ) y \relax (0)+\left (x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {1}{60} x^{5}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 70
AsymptoticDSolveValue[(x^2-1)*y''[x]+(1-x)*y'[x]+(x^2-2*x+1)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (-\frac {x^5}{15}+\frac {x^4}{12}-\frac {x^3}{6}+\frac {x^2}{2}+1\right )+c_2 \left (\frac {x^5}{60}-\frac {x^4}{12}+\frac {x^3}{6}+\frac {x^2}{2}+x\right ) \]