82.3.24 problem 25-24

Internal problem ID [21807]
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 : 25-24
Date solved : Thursday, October 02, 2025 at 08:02:27 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 48

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

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+2 x +x^{2}+\frac {2}{9} x^{3}+\frac {1}{36} x^{4}+\frac {1}{450} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-4\right ) x -3 x^{2}-\frac {22}{27} x^{3}-\frac {25}{216} x^{4}-\frac {137}{13500} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2}{x} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 112

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

\[ y(x)\to \frac {c_1 \left (\frac {x^5}{450}+\frac {x^4}{36}+\frac {2 x^3}{9}+x^2+2 x+1\right )}{x}+c_2 \left (\frac {-\frac {137 x^5}{13500}-\frac {25 x^4}{216}-\frac {22 x^3}{27}-3 x^2-4 x}{x}+\frac {\left (\frac {x^5}{450}+\frac {x^4}{36}+\frac {2 x^3}{9}+x^2+2 x+1\right ) \log (x)}{x}\right ) \]

✓ Sympy. Time used: 0.281 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = \frac {C_{1} \left (\frac {x^{6}}{8100} + \frac {x^{5}}{450} + \frac {x^{4}}{36} + \frac {2 x^{3}}{9} + x^{2} + 2 x + 1\right )}{x} + O\left (x^{6}\right ) \]