82.3.6 problem 25-6

Internal problem ID [21789]
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 : 25-6
Date solved : Thursday, October 02, 2025 at 08:02:14 PM
CAS classification : [_Jacobi]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 26

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

\[ y = c_1 \sqrt {x}\, \left (1+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+x +\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 18

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

\[ y(x)\to c_1 \sqrt {x}+c_2 (x+1) \]

✓ Sympy. Time used: 0.365 (sec). Leaf size: 14

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

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