Internal problem ID [2457]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition,
2015
Section: Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.5. page
771
Problem number: 24.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+2 x \left (2+x \right ) y^{\prime }+2 \left (x +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 51
Order:=6; dsolve(x^2*diff(y(x),x$2)+2*x*(2+x)*diff(y(x),x)+2*(1+x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} \left (1+\mathrm {O}\left (x^{6}\right )\right ) x +\left (2 x +\mathrm {O}\left (x^{6}\right )\right ) \ln \relax (x ) c_{2}+\left (1-2 x -2 x^{2}+\frac {2}{3} x^{3}-\frac {2}{9} x^{4}+\frac {1}{15} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.031 (sec). Leaf size: 48
AsymptoticDSolveValue[x^2*y''[x]+2*x*(2+x)*y'[x]+2*(1+x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {2 \log (x)}{x}-\frac {2 x^4-6 x^3+18 x^2+36 x-9}{9 x^2}\right )+\frac {c_2}{x} \]