49.15.10 problem 6

Internal problem ID [7696]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 3. Linear equations with variable coefficients. Page 130
Problem number : 6
Date solved : Wednesday, March 05, 2025 at 04:50:40 AM
CAS classification : [_Gegenbauer]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 81

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 127

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

\[ y(x)\to c_2 \left (\frac {1}{60} \left (-\alpha ^2-\alpha \right ) x^5-\frac {1}{120} \left (-\alpha ^2-\alpha \right ) \left (\alpha ^2+\alpha \right ) x^5-\frac {1}{10} \left (\alpha ^2+\alpha \right ) x^5+\frac {x^5}{5}-\frac {1}{6} \left (\alpha ^2+\alpha \right ) x^3+\frac {x^3}{3}+x\right )+c_1 \left (\frac {1}{24} \left (\alpha ^2+\alpha \right )^2 x^4-\frac {1}{4} \left (\alpha ^2+\alpha \right ) x^4-\frac {1}{2} \left (\alpha ^2+\alpha \right ) x^2+1\right ) \]

✓ Sympy. Time used: 1.139 (sec). Leaf size: 83

from sympy import * 
x = symbols("x") 
Alpha = symbols("Alpha") 
y = Function("y") 
ode = Eq(Alpha*(Alpha + 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)
 

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