87.26.7 problem 7

Internal problem ID [23841]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 5. Series solutions of second order linear equations. Exercise at page 253
Problem number : 7
Date solved : Thursday, October 02, 2025 at 09:45:43 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }+\frac {y}{16}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 69

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*(1-x)*diff(y(x),x)+1/16*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{-\frac {i}{4}} \left (1+\left (\frac {1}{10}-\frac {i}{5}\right ) x +\left (\frac {1}{40}-\frac {i}{20}\right ) x^{2}+\left (\frac {53}{8880}-\frac {2 i}{185}\right ) x^{3}+\left (\frac {17}{14208}-\frac {7 i}{3552}\right ) x^{4}+\left (\frac {581}{2870016}-\frac {2203 i}{7175040}\right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \,x^{\frac {i}{4}} \left (1+\left (\frac {1}{10}+\frac {i}{5}\right ) x +\left (\frac {1}{40}+\frac {i}{20}\right ) x^{2}+\left (\frac {53}{8880}+\frac {2 i}{185}\right ) x^{3}+\left (\frac {17}{14208}+\frac {7 i}{3552}\right ) x^{4}+\left (\frac {581}{2870016}+\frac {2203 i}{7175040}\right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 134

ode=x^2*D[y[x],{x,2}]+x*(1-x)*D[y[x],x]+1/16*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 x^{1 i/4} \left (\left (\frac {581}{2870016}+\frac {2203 i}{7175040}\right ) x^5+\left (\frac {17}{14208}+\frac {7 i}{3552}\right ) x^4+\left (\frac {53}{8880}+\frac {2 i}{185}\right ) x^3+\left (\frac {1}{40}+\frac {i}{20}\right ) x^2+\left (\frac {1}{10}+\frac {i}{5}\right ) x+1\right )+c_2 x^{-i/4} \left (\left (\frac {581}{2870016}-\frac {2203 i}{7175040}\right ) x^5+\left (\frac {17}{14208}-\frac {7 i}{3552}\right ) x^4+\left (\frac {53}{8880}-\frac {2 i}{185}\right ) x^3+\left (\frac {1}{40}-\frac {i}{20}\right ) x^2+\left (\frac {1}{10}-\frac {i}{5}\right ) x+1\right ) \]

✗ 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)/16,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : Expected Expr or iterable but got None