82.3.1 problem 25-1

Internal problem ID [21784]
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 : 25-1
Date solved : Thursday, October 02, 2025 at 08:02:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 58

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

\[ y = c_1 x \left (1-\frac {1}{4} x -\frac {1}{48} x^{2}-\frac {17}{1152} x^{3}+\frac {1157}{46080} x^{4}+\frac {63863}{2764800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{96} x^{3}+\frac {17}{2304} x^{4}-\frac {1157}{92160} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {1}{16} x^{2}+\frac {29}{288} x^{3}+\frac {3617}{27648} x^{4}+\frac {317857}{2764800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 87

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

\[ y(x)\to c_1 \left (\frac {x \left (17 x^3+24 x^2+288 x-1152\right ) \log (x)}{2304}+\frac {4025 x^4+3360 x^3+5184 x^2-27648 x+27648}{27648}\right )+c_2 \left (\frac {1157 x^5}{46080}-\frac {17 x^4}{1152}-\frac {x^3}{48}-\frac {x^2}{4}+x\right ) \]

✓ Sympy. Time used: 0.607 (sec). Leaf size: 10

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

\[ y{\left (x \right )} = C_{2} x + C_{1} + O\left (x^{6}\right ) \]