82.2.10 problem 24-10

Internal problem ID [21771]
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 : 24-10
Date solved : Thursday, October 02, 2025 at 08:01:58 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} \left (x^{2}+2\right ) y^{\prime \prime }+\left (2 x +\frac {2}{x}\right ) y^{\prime }+2 x^{2} y&=\frac {4 x^{2}+2 x +10}{x^{4}} \end{align*}

Using series method with expansion around

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

✗ Maple

Order:=6; 
ode:=(x^2+2)*diff(diff(y(x),x),x)+(2*x+2/x)*diff(y(x),x)+2*x^2*y(x) = (4*x^2+2*x+10)/x^4; 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.109 (sec). Leaf size: 296

ode=(x^2+2)*D[y[x],{x,2}]+(2*x+2/x)*D[y[x],x]+2*x^2*y[x]==(4*x^2+2*x+10)/x^4; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {5 x^6}{288}-\frac {x^4}{16}+1\right )+c_1 \left (x^6 \left (\frac {5 \log (x)}{144}-\frac {7}{3456}\right )+x^4 \left (\frac {3}{64}-\frac {\log (x)}{8}\right )-\frac {x^2}{4}+2 \log (x)+1\right )+\left (\frac {413 x^6}{110592}-\frac {7 x^5}{11520}+\frac {37 x^4}{9216}+\frac {x^3}{192}-\frac {11 x^2}{128}-\frac {5}{4 x^2}-\frac {x}{8}-\frac {1}{2 x}+\frac {3 \log (x)}{8}\right ) \left (x^6 \left (\frac {5 \log (x)}{144}-\frac {7}{3456}\right )+x^4 \left (\frac {3}{64}-\frac {\log (x)}{8}\right )-\frac {x^2}{4}+2 \log (x)+1\right )+\left (\frac {5 x^6}{288}-\frac {x^4}{16}+1\right ) \left (-\frac {59 x^6 (84 \log (x)+13)}{663552}+\frac {x^5 (420 \log (x)+2551)}{345600}+\frac {x^4 (-222 \log (x)-31)}{27648}+\frac {x^3 (-12 \log (x)-35)}{1152}+\frac {1}{256} x^2 (44 \log (x)-23)+\frac {5 (\log (x)+1)}{2 x^2}+\frac {1}{4} x \log (x)-\frac {1}{8} \log (x) (3 \log (x)-2)+\frac {2 \log (x)+3}{2 x}\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*y(x) + (2*x + 2/x)*Derivative(y(x), x) + (x**2 + 2)*Derivative(y(x), (x, 2)) - (4*x**2 + 2*x + 10)/x**4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE 2*x**2*y(x) + (2*x + 2/x)*Derivative(y(x), x) + (x**2 + 2)*Derivative(y(x), (x, 2)) - (4*x**2 + 2*x + 10)/x**4 does not match hint 2nd_power_series_regular