82.3.2 problem 25-2

Internal problem ID [21785]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 25. Power series about a singular point. Page 762
Problem number : 25-2
Date solved : Thursday, October 02, 2025 at 08:02:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 370

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

\[ y = \frac {c_2 \left (x -1\right )^{\frac {i \sqrt {3}}{2}} \left (1+\frac {-4 \sqrt {3}+5 i}{2 i-2 \sqrt {3}} \left (x -1\right )+\frac {3 \sqrt {3}-\frac {9 i}{8}}{\left (-i+\sqrt {3}\right ) \left (i \sqrt {3}+2\right )} \left (x -1\right )^{2}+\frac {1}{48} \frac {100 i \sqrt {3}+197}{\left (-\sqrt {3}+2 i\right ) \left (-i+\sqrt {3}\right ) \left (i \sqrt {3}+3\right )} \left (x -1\right )^{3}+\frac {\frac {25 \sqrt {3}}{48}+\frac {3145 i}{384}}{\left (i-\sqrt {3}\right ) \left (i \sqrt {3}+2\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+4\right )} \left (x -1\right )^{4}+\frac {1}{256} \frac {8140 \sqrt {3}+3363 i}{\left (i-\sqrt {3}\right ) \left (i \sqrt {3}+2\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+5\right )} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right )+c_1 \left (x -1\right )^{-\frac {i \sqrt {3}}{2}} \left (1+\frac {5 i+4 \sqrt {3}}{2 \sqrt {3}+2 i} \left (x -1\right )+\frac {-9 i-24 \sqrt {3}}{8 i-24 \sqrt {3}} \left (x -1\right )^{2}+\frac {-197 i-100 \sqrt {3}}{576 i-384 \sqrt {3}} \left (x -1\right )^{3}+\frac {\frac {25 i \sqrt {3}}{48}+\frac {3145}{384}}{\left (\sqrt {3}+i\right ) \left (\sqrt {3}+2 i\right ) \left (\sqrt {3}+3 i\right ) \left (\sqrt {3}+4 i\right )} \left (x -1\right )^{4}+\frac {8140 \sqrt {3}-3363 i}{107520 i-7168 \sqrt {3}} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right )}{\sqrt {x -1}} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 3398

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

Too large to display

✗ Sympy

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

ValueError : Expected Expr or iterable but got None