50.18.13 problem 7

Internal problem ID [8107]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.3. Second-Order Linear Equations: Ordinary Points. Page 169
Problem number : 7
Date solved : Wednesday, March 05, 2025 at 05:29:58 AM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 103

Order:=8; 
ode:=(-x^2+1)*diff(diff(y(x),x),x)-x*diff(y(x),x)+p^2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {p^{2} x^{2}}{2}+\frac {p^{2} \left (p^{2}-4\right ) x^{4}}{24}-\frac {p^{2} \left (p^{4}-20 p^{2}+64\right ) x^{6}}{720}\right ) y \left (0\right )+\left (x -\frac {\left (p^{2}-1\right ) x^{3}}{6}+\frac {\left (p^{4}-10 p^{2}+9\right ) x^{5}}{120}-\frac {\left (p^{6}-35 p^{4}+259 p^{2}-225\right ) x^{7}}{5040}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 155

ode=(1-x^2)*D[y[x],{x,2}]-x*D[y[x],x]+p^2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (-\frac {p^6 x^7}{5040}+\frac {p^4 x^7}{144}+\frac {p^4 x^5}{120}-\frac {37 p^2 x^7}{720}-\frac {p^2 x^5}{12}-\frac {p^2 x^3}{6}+\frac {5 x^7}{112}+\frac {3 x^5}{40}+\frac {x^3}{6}+x\right )+c_1 \left (-\frac {1}{720} p^6 x^6+\frac {p^4 x^6}{36}+\frac {p^4 x^4}{24}-\frac {4 p^2 x^6}{45}-\frac {p^2 x^4}{6}-\frac {p^2 x^2}{2}+1\right ) \]

✓ Sympy. Time used: 1.101 (sec). Leaf size: 104

from sympy import * 
x = symbols("x") 
p = symbols("p") 
y = Function("y") 
ode = Eq(p**2*y(x) - 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=8)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {p^{6} x^{6}}{720} + \frac {p^{4} x^{6}}{36} + \frac {p^{4} x^{4}}{24} - \frac {4 p^{2} x^{6}}{45} - \frac {p^{2} x^{4}}{6} - \frac {p^{2} x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {p^{4} x^{4}}{120} - \frac {p^{2} x^{4}}{12} - \frac {p^{2} x^{2}}{6} + \frac {3 x^{4}}{40} + \frac {x^{2}}{6} + 1\right ) + O\left (x^{8}\right ) \]