82.3.25 problem 25-25

Internal problem ID [21808]
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 : 25-25
Date solved : Thursday, October 02, 2025 at 08:02:28 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 46

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

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {1}{4} x^{2}+\frac {1}{18} x^{3}+\frac {1}{192} x^{4}-\frac {11}{3600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-x +\frac {1}{2} x^{2}-\frac {1}{108} x^{3}-\frac {41}{1152} x^{4}+\frac {1529}{216000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2}{x} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 116

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

\[ y(x)\to \frac {c_1 \left (-\frac {11 x^5}{3600}+\frac {x^4}{192}+\frac {x^3}{18}-\frac {x^2}{4}+1\right )}{x}+c_2 \left (\frac {\frac {1529 x^5}{216000}-\frac {41 x^4}{1152}-\frac {x^3}{108}+\frac {x^2}{2}-x}{x}+\frac {\left (-\frac {11 x^5}{3600}+\frac {x^4}{192}+\frac {x^3}{18}-\frac {x^2}{4}+1\right ) \log (x)}{x}\right ) \]

✓ Sympy. Time used: 0.334 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = \frac {C_{1} \left (\frac {29 x^{6}}{103680} - \frac {11 x^{5}}{3600} + \frac {x^{4}}{192} + \frac {x^{3}}{18} - \frac {x^{2}}{4} + 1\right )}{x} + O\left (x^{6}\right ) \]