87.26.6 problem 6

Internal problem ID [23840]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 5. Series solutions of second order linear equations. Exercise at page 253
Problem number : 6
Date solved : Thursday, October 02, 2025 at 09:45:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+4 x \left (1-x \right ) y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 50

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+4*x*(1-x)*diff(y(x),x)+2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {\left (\left (-8\right ) x +16 x^{2}+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_2 +c_1 \left (1-2 x +\operatorname {O}\left (x^{6}\right )\right ) x +\left (1+4 x -48 x^{2}+\frac {32}{3} x^{3}+\frac {32}{9} x^{4}+\frac {64}{45} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2}{x^{2}} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 55

ode=x^2*D[y[x],{x,2}]+4*x*(1-x)*D[y[x],x]+2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {32 x^4+96 x^3-576 x^2+108 x+9}{9 x^2}+\frac {8 (2 x-1) \log (x)}{x}\right )+c_2 \left (\frac {1}{x}-2\right ) \]

✓ Sympy. Time used: 0.269 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + 4*x*(1 - x)*Derivative(y(x), x) + 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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