Internal problem ID [2432]
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: (b).
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 }-\left (-1+2 \sqrt {5}\right ) x y^{\prime }+\left (\frac {19}{4}-3 x^{2}\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.037 (sec). Leaf size: 325
Order:=6; dsolve(x^2*diff(y(x),x$2)-(2*sqrt(5)-1)*x*diff(y(x),x)+(19/4-3*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = x^{\frac {1}{2}+\sqrt {5}} \left (1+\frac {1}{2} x^{2}+\frac {3}{40} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) \ln \relax (x ) c_{2}+c_{1} x^{-\frac {1}{2}+\sqrt {5}} \left (1+\frac {3}{2} x^{2}+\frac {3}{8} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+x^{\frac {1}{2}+\sqrt {5}} \left (-\frac {5}{12} x^{2}-\frac {77}{800} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.057 (sec). Leaf size: 94
AsymptoticDSolveValue[x^2*y''[x]-(2*Sqrt[5]-1)*x*y'[x]+(19/4-3*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {3}{8} x^{\frac {7}{2}+\sqrt {5}}+\frac {3}{2} x^{\frac {3}{2}+\sqrt {5}}+x^{\sqrt {5}-\frac {1}{2}}\right )+c_2 \left (\frac {3}{40} x^{\frac {9}{2}+\sqrt {5}}+\frac {1}{2} x^{\frac {5}{2}+\sqrt {5}}+x^{\frac {1}{2}+\sqrt {5}}\right ) \]