20.26.5 problem (b)

Internal problem ID [4030]
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)
Date solved : Sunday, March 30, 2025 at 02:15:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-\left (2 \sqrt {5}-1\right ) x y^{\prime }+\left (\frac {19}{4}-3 x^{2}\right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.049 (sec). Leaf size: 50

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)-(-1+2*5^(1/2))*x*diff(y(x),x)+(19/4-3*x^2)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = x^{-\frac {1}{2}+\sqrt {5}} \left (\left (1+\frac {3}{2} x^{2}+\frac {3}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_1 +x c_2 \left (\left (1+\frac {1}{2} x^{2}+\frac {3}{40} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right )+\left (-\frac {5}{12} x^{2}-\frac {77}{800} x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) \]

✓ Mathematica. Time used: 0.059 (sec). Leaf size: 94

ode=x^2*D[y[x],{x,2}]-(2*Sqrt[5]-1)*x*D[y[x],x]+(19/4-3*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {3}{8} x^{\frac {7}{2}+\sqrt {5}}+\frac {3}{2} x^{\frac {3}{2}+\sqrt {5}}+x^{\sqrt {5}-\frac {1}{2}}\right )+c_2 \left (\frac {3}{40} x^{\frac {9}{2}+\sqrt {5}}+\frac {1}{2} x^{\frac {5}{2}+\sqrt {5}}+x^{\frac {1}{2}+\sqrt {5}}\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - x*(-1 + 2*sqrt(5))*Derivative(y(x), x) + (19/4 - 3*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

NotImplementedError : Not sure of sign of 6 - x1