50.22.15 problem 2(g)

Internal problem ID [8158]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Problems for review and discovert. (A) Drill Exercises . Page 194
Problem number : 2(g)
Date solved : Sunday, March 30, 2025 at 12:46:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 55

Order:=8; 
ode:=x^2*diff(diff(y(x),x),x)+x*(1-x)*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{-i} \left (1+\left (\frac {2}{5}-\frac {i}{5}\right ) x +\left (\frac {1}{10}-\frac {i}{20}\right ) x^{2}+\left (\frac {17}{780}-\frac {i}{130}\right ) x^{3}+\left (\frac {5}{1248}-\frac {i}{1248}\right ) x^{4}+\left (\frac {113}{180960}-\frac {7 i}{180960}\right ) x^{5}+\left (\frac {911}{10857600}+\frac {19 i}{3619200}\right ) x^{6}+\left (\frac {39799}{4028169600}+\frac {1009 i}{575452800}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \,x^{i} \left (1+\left (\frac {2}{5}+\frac {i}{5}\right ) x +\left (\frac {1}{10}+\frac {i}{20}\right ) x^{2}+\left (\frac {17}{780}+\frac {i}{130}\right ) x^{3}+\left (\frac {5}{1248}+\frac {i}{1248}\right ) x^{4}+\left (\frac {113}{180960}+\frac {7 i}{180960}\right ) x^{5}+\left (\frac {911}{10857600}-\frac {19 i}{3619200}\right ) x^{6}+\left (\frac {39799}{4028169600}-\frac {1009 i}{575452800}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 122

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

\[ y(x)\to \left (\frac {59}{10857600}-\frac {17 i}{10857600}\right ) c_2 x^{-i} \left ((14+5 i) x^6+(108+24 i) x^5+(720+60 i) x^4+(4080-240 i) x^3+(19440-3600 i) x^2+(77760-14400 i) x+(169920+48960 i)\right )+\left (\frac {59}{10857600}+\frac {17 i}{10857600}\right ) c_1 x^i \left ((14-5 i) x^6+(108-24 i) x^5+(720-60 i) x^4+(4080+240 i) x^3+(19440+3600 i) x^2+(77760+14400 i) x+(169920-48960 i)\right ) \]

✗ 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) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

ValueError : Expected Expr or iterable but got None