82.2.22 problem 24-23

Internal problem ID [21783]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 24. Power series about an ordinary point. Page 719
Problem number : 24-23
Date solved : Thursday, October 02, 2025 at 08:02:06 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 74

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

\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{2} x^{3}-\frac {1}{4} x^{4}+\frac {1}{8} x^{5}\right ) y \left (0\right )+\left (x -\frac {3}{2} x^{2}+\frac {4}{3} x^{3}-\frac {7}{8} x^{4}+\frac {61}{120} x^{5}\right ) y^{\prime }\left (0\right )+\frac {x^{3}}{6}-\frac {7 x^{4}}{24}+\frac {4 x^{5}}{15}+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 91

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

\[ y(x)\to \frac {4 x^5}{15}-\frac {7 x^4}{24}+\frac {x^3}{6}+c_1 \left (\frac {x^5}{8}-\frac {x^4}{4}+\frac {x^3}{2}-\frac {x^2}{2}+1\right )+c_2 \left (\frac {61 x^5}{120}-\frac {7 x^4}{8}+\frac {4 x^3}{3}-\frac {3 x^2}{2}+x\right ) \]

✗ Sympy

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

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