67.26.12 problem 36.2 (L)

Internal problem ID [17039]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 36. The big theorem on the the Frobenius method. Additional Exercises. page 739
Problem number : 36.2 (L)
Date solved : Thursday, October 02, 2025 at 01:42:13 PM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 46

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

\[ y = c_1 \sqrt {x -1}\, \left (1+\frac {11}{12} \left (x -1\right )+\frac {11}{160} \left (x -1\right )^{2}-\frac {143}{13440} \left (x -1\right )^{3}+\frac {5291}{1935360} \left (x -1\right )^{4}-\frac {11063}{12902400} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right )+c_2 \left (1+3 \left (x -1\right )+\left (x -1\right )^{2}-\frac {1}{15} \left (x -1\right )^{3}+\frac {1}{70} \left (x -1\right )^{4}-\frac {13}{3150} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 101

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

\[ y(x)\to c_1 \left (-\frac {11063 (x-1)^5}{12902400}+\frac {5291 (x-1)^4}{1935360}-\frac {143 (x-1)^3}{13440}+\frac {11}{160} (x-1)^2+\frac {11 (x-1)}{12}+1\right ) \sqrt {x-1}+c_2 \left (-\frac {13 (x-1)^5}{3150}+\frac {1}{70} (x-1)^4-\frac {1}{15} (x-1)^3+(x-1)^2+3 (x-1)+1\right ) \]

✗ Sympy

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

PolynomialError : non-commutative expressions are not supported