78.22.1 problem 1 (a)

Internal problem ID [18371]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 5. Power Series Solutions and Special Functions. Section 32. The point at infinity. Problems at page 244
Problem number : 1 (a)
Date solved : Thursday, March 13, 2025 at 11:55:08 AM
CAS classification : [_Gegenbauer]

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

Using series method with expansion around

\begin{align*} \infty \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 142

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

\[ y = \frac {c_{1} \left (\frac {1}{x}\right )^{n} \left (1+\frac {n^{2}+3 n +2}{2 \left (2 n +3\right ) x^{2}}+\frac {n^{4}+10 n^{3}+35 n^{2}+50 n +24}{8 \left (2 n +5\right ) \left (2 n +3\right ) x^{4}}+O\left (\frac {1}{x^{6}}\right )\right )}{x}+c_{2} \left (\frac {1}{x}\right )^{-n} \left (1-\frac {n \left (n -1\right )}{2 \left (2 n -1\right ) x^{2}}+\frac {n \left (n^{3}-6 n^{2}+11 n -6\right )}{8 \left (2 n -3\right ) \left (2 n -1\right ) x^{4}}+O\left (\frac {1}{x^{6}}\right )\right ) \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 759

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

\[ y(x)\to c_1 \left (\frac {n^2 x^{-n-5}}{-n^2-n+(n+4) (n+5)}+\frac {3 n x^{-n-5}}{-n^2-n+(n+4) (n+5)}+\frac {17 n^2 x^{-n-5}}{\left (-n^2-n+(n+2) (n+3)\right ) \left (-n^2-n+(n+4) (n+5)\right )}+\frac {24 n x^{-n-5}}{\left (-n^2-n+(n+2) (n+3)\right ) \left (-n^2-n+(n+4) (n+5)\right )}+\frac {12 x^{-n-5}}{\left (-n^2-n+(n+2) (n+3)\right ) \left (-n^2-n+(n+4) (n+5)\right )}+\frac {2 x^{-n-5}}{-n^2-n+(n+4) (n+5)}+\frac {n^2 x^{-n-3}}{-n^2-n+(n+2) (n+3)}+\frac {3 n x^{-n-3}}{-n^2-n+(n+2) (n+3)}+\frac {2 x^{-n-3}}{-n^2-n+(n+2) (n+3)}+\frac {n^4 x^{-n-5}}{\left (-n^2-n+(n+2) (n+3)\right ) \left (-n^2-n+(n+4) (n+5)\right )}+\frac {6 n^3 x^{-n-5}}{\left (-n^2-n+(n+2) (n+3)\right ) \left (-n^2-n+(n+4) (n+5)\right )}+x^{-n-1}\right )+c_2 \left (\frac {n^2 x^{n-4}}{-n^2-n+(3-n) (4-n)}-\frac {n x^{n-4}}{-n^2-n+(3-n) (4-n)}+\frac {5 n^2 x^{n-4}}{\left (-n^2-n+(1-n) (2-n)\right ) \left (-n^2-n+(3-n) (4-n)\right )}-\frac {4 n x^{n-4}}{\left (-n^2-n+(1-n) (2-n)\right ) \left (-n^2-n+(3-n) (4-n)\right )}+\frac {n^2 x^{n-2}}{-n^2-n+(1-n) (2-n)}-\frac {n x^{n-2}}{-n^2-n+(1-n) (2-n)}+\frac {n^4 x^{n-4}}{\left (-n^2-n+(1-n) (2-n)\right ) \left (-n^2-n+(3-n) (4-n)\right )}-\frac {2 n^3 x^{n-4}}{\left (-n^2-n+(1-n) (2-n)\right ) \left (-n^2-n+(3-n) (4-n)\right )}+x^n\right ) \]

✓ Sympy. Time used: 3.954 (sec). Leaf size: 382

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

\[ y{\left (x \right )} = C_{3} \left (\frac {C_{2}^{2} n^{2} \left (C_{2} - x\right )^{4}}{C_{2}^{6} - 3 C_{2}^{4} + 3 C_{2}^{2} - 1} + \frac {C_{2}^{2} n \left (C_{2} - x\right )^{4}}{C_{2}^{6} - 3 C_{2}^{4} + 3 C_{2}^{2} - 1} + \frac {2 C_{2} n^{2} \left (C_{2} - x\right )^{3}}{3 \left (C_{2}^{4} - 2 C_{2}^{2} + 1\right )} + \frac {2 C_{2} n \left (C_{2} - x\right )^{3}}{3 \left (C_{2}^{4} - 2 C_{2}^{2} + 1\right )} + \frac {n^{4} \left (C_{2} - x\right )^{4}}{24 \left (C_{2}^{4} - 2 C_{2}^{2} + 1\right )} + \frac {n^{3} \left (C_{2} - x\right )^{4}}{12 \left (C_{2}^{4} - 2 C_{2}^{2} + 1\right )} - \frac {5 n^{2} \left (C_{2} - x\right )^{4}}{24 \left (C_{2}^{4} - 2 C_{2}^{2} + 1\right )} + \frac {n^{2} \left (C_{2} - x\right )^{2}}{2 \left (C_{2}^{2} - 1\right )} - \frac {n \left (C_{2} - x\right )^{4}}{4 \left (C_{2}^{4} - 2 C_{2}^{2} + 1\right )} + \frac {n \left (C_{2} - x\right )^{2}}{2 \left (C_{2}^{2} - 1\right )} + 1\right ) + C_{1} \left (- \frac {2 C_{2}^{3} \left (C_{2} - x\right )^{4}}{C_{2}^{6} - 3 C_{2}^{4} + 3 C_{2}^{2} - 1} - \frac {4 C_{2}^{2} \left (C_{2} - x\right )^{3}}{3 \left (C_{2}^{4} - 2 C_{2}^{2} + 1\right )} - \frac {C_{2} n^{2} \left (C_{2} - x\right )^{4}}{3 \left (C_{2}^{4} - 2 C_{2}^{2} + 1\right )} - \frac {C_{2} n \left (C_{2} - x\right )^{4}}{3 \left (C_{2}^{4} - 2 C_{2}^{2} + 1\right )} + \frac {C_{2} \left (C_{2} - x\right )^{4}}{C_{2}^{4} - 2 C_{2}^{2} + 1} - \frac {C_{2} \left (C_{2} - x\right )^{2}}{C_{2}^{2} - 1} - C_{2} - \frac {n^{2} \left (C_{2} - x\right )^{3}}{6 \left (C_{2}^{2} - 1\right )} - \frac {n \left (C_{2} - x\right )^{3}}{6 \left (C_{2}^{2} - 1\right )} + x + \frac {\left (C_{2} - x\right )^{3}}{3 \left (C_{2}^{2} - 1\right )}\right ) + O\left (x^{6}\right ) \]