20.25.5 problem 6

Internal problem ID [4010]
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 11.4. page 758
Problem number : 6
Date solved : Tuesday, March 04, 2025 at 05:22:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-7 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 213

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*(1-x)*diff(y(x),x)-7*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y \left (x \right ) = c_{1} x^{-\sqrt {7}} \left (1+\frac {\sqrt {7}}{-1+2 \sqrt {7}} x +\frac {\sqrt {7}}{-4+8 \sqrt {7}} x^{2}+\frac {\sqrt {7}\, \left (\sqrt {7}-2\right )}{372-96 \sqrt {7}} x^{3}+\frac {\sqrt {7}\, \left (\sqrt {7}-3\right )}{2976-768 \sqrt {7}} x^{4}+\frac {\left (\sqrt {7}-4\right ) \left (\sqrt {7}-3\right ) \sqrt {7}}{48960 \sqrt {7}-128160} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\sqrt {7}} \left (1+\frac {\sqrt {7}}{1+2 \sqrt {7}} x +\frac {\sqrt {7}}{4+8 \sqrt {7}} x^{2}+\frac {\sqrt {7}\, \left (\sqrt {7}+2\right )}{372+96 \sqrt {7}} x^{3}+\frac {\left (\sqrt {7}+3\right ) \sqrt {7}}{2976+768 \sqrt {7}} x^{4}+\frac {\left (\sqrt {7}+4\right ) \left (\sqrt {7}+3\right ) \sqrt {7}}{48960 \sqrt {7}+128160} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 1066

ode=x^2*D[y[x],{x,2}]+x*(1-x)*D[y[x],x]-7*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(1 - x)*Derivative(y(x), x) - 7*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 - x0