10.14.13 problem 10

Internal problem ID [1395]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 5.3, Series Solutions Near an Ordinary Point, Part II. page 269
Problem number : 10
Date solved : Tuesday, March 04, 2025 at 12:35:08 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 }+\alpha ^{2} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 65

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

\[ y = \left (1-\frac {\alpha ^{2} x^{2}}{2}+\frac {\alpha ^{2} \left (\alpha ^{2}-4\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (\alpha ^{2}-1\right ) x^{3}}{6}+\frac {\left (\alpha ^{4}-10 \alpha ^{2}+9\right ) x^{5}}{120}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 88

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

\[ y(x)\to c_2 \left (\frac {\alpha ^4 x^5}{120}-\frac {\alpha ^2 x^5}{12}+\frac {3 x^5}{40}-\frac {\alpha ^2 x^3}{6}+\frac {x^3}{6}+x\right )+c_1 \left (\frac {\alpha ^4 x^4}{24}-\frac {\alpha ^2 x^4}{6}-\frac {\alpha ^2 x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.964 (sec). Leaf size: 53

from sympy import * 
x = symbols("x") 
Alpha = symbols("Alpha") 
y = Function("y") 
ode = Eq(Alpha**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=6)
 

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