28.3.15 problem 18

Internal problem ID [7189]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XIV. page 177
Problem number : 18
Date solved : Tuesday, September 30, 2025 at 04:25:13 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 }+a^{2} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 65

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

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

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

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

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

✓ Sympy. Time used: 0.338 (sec). Leaf size: 53

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

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