73.25.19 problem 35.4 (e)

Internal problem ID [15656]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional Exercises. page 715
Problem number : 35.4 (e)
Date solved : Monday, March 31, 2025 at 01:44:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x y^{\prime }+\frac {y}{1-x}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 38

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

\[ y = \left (\left (2 x -\frac {1}{2} x^{2}-\frac {1}{6} x^{3}-\frac {1}{12} x^{4}-\frac {1}{20} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1-x +\operatorname {O}\left (x^{6}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right )\right ) x \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 60

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

\[ y(x)\to c_2 \left (x \left (-\frac {x^5}{20}-\frac {x^4}{12}-\frac {x^3}{6}-\frac {x^2}{2}+2 x\right )+(1-x) x \log (x)\right )+c_1 (1-x) x \]

✓ Sympy. Time used: 0.934 (sec). Leaf size: 8

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

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