Internal problem ID [2441]
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: 6.
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 }+y^{\prime } x +\left (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: 0.031 (sec). Leaf size: 63
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(2*x-1)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x^{2} \left (1-\frac {2}{3} x +\frac {1}{6} x^{2}-\frac {1}{45} x^{3}+\frac {1}{540} x^{4}-\frac {1}{9450} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (4 x^{2}-\frac {8}{3} x^{3}+\frac {2}{3} x^{4}-\frac {4}{45} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (-2-4 x +\frac {32}{9} x^{3}-\frac {25}{18} x^{4}+\frac {157}{675} x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.021 (sec). Leaf size: 83
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]+(2*x-1)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {31 x^4-88 x^3+36 x^2+72 x+36}{36 x}-\frac {1}{3} x \left (x^2-4 x+6\right ) \log (x)\right )+c_2 \left (\frac {x^5}{540}-\frac {x^4}{45}+\frac {x^3}{6}-\frac {2 x^2}{3}+x\right ) \]