40.16.9 problem 16

Internal problem ID [6800]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 25. Integration in series. Supplemetary problems. Page 205
Problem number : 16
Date solved : Wednesday, March 05, 2025 at 02:46:06 AM
CAS classification : [_Gegenbauer]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 81

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

\[ y = \left (1-\frac {p \left (p +1\right ) x^{2}}{2}+\frac {p \left (p^{3}+2 p^{2}-5 p -6\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (p^{2}+p -2\right ) x^{3}}{6}+\frac {\left (p^{4}+2 p^{3}-13 p^{2}-14 p +24\right ) x^{5}}{120}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 120

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

\[ y(x)\to c_2 \left (\frac {1}{120} \left (p^2+p\right )^2 x^5+\frac {7}{60} \left (-p^2-p\right ) x^5+\frac {1}{6} \left (-p^2-p\right ) x^3+\frac {x^5}{5}+\frac {x^3}{3}+x\right )+c_1 \left (\frac {1}{24} \left (p^2+p\right )^2 x^4+\frac {1}{4} \left (-p^2-p\right ) x^4+\frac {1}{2} \left (-p^2-p\right ) x^2+1\right ) \]

✓ Sympy. Time used: 1.235 (sec). Leaf size: 83

from sympy import * 
x = symbols("x") 
p = symbols("p") 
y = Function("y") 
ode = Eq(p*(p + 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=0,n=6)
 

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