1.15.16 problem 16

Internal problem ID [472]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 16
Date solved : Tuesday, September 30, 2025 at 03:59:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 80

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

\[ y(x)\to c_1 \left (-\frac {3 x^4}{16}+\frac {x^3}{4}-\frac {x^2}{4}+1\right )+\frac {c_2 e^{\frac {1}{x^2}+\frac {5}{x}} \left (-\frac {8921 x^5}{16}+\frac {1629 x^4}{8}-\frac {311 x^3}{4}+28 x^2-\frac {15 x}{2}+1\right )}{x^2} \]

✗ Sympy

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

ValueError : ODE x**3*(1 - x)*Derivative(y(x), (x, 2)) + x*y(x) + (3*x + 2)*Derivative(y(x), x) does not match hint 2nd_power_series_regular