Internal problem ID [4501]
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: 8.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {{\mathrm e}^{x} y^{\prime \prime }-\left (x^{2}-1\right ) y^{\prime }+2 x y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 49
Order:=6; dsolve(exp(x)*diff(y(x),x$2)-(x^2-1)*diff(y(x),x)+2*x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {1}{3} x^{3}+\frac {1}{4} x^{4}-\frac {3}{20} x^{5}\right ) y \relax (0)+\left (x -\frac {1}{2} x^{2}+\frac {1}{3} x^{3}-\frac {7}{24} x^{4}+\frac {23}{120} x^{5}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 63
AsymptoticDSolveValue[Exp[x]*y''[x]-(x^2-1)*y'[x]+2*x*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (-\frac {3 x^5}{20}+\frac {x^4}{4}-\frac {x^3}{3}+1\right )+c_2 \left (\frac {23 x^5}{120}-\frac {7 x^4}{24}+\frac {x^3}{3}-\frac {x^2}{2}+x\right ) \]