28.3.14 problem 16

Internal problem ID [7188]
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 : 16
Date solved : Tuesday, September 30, 2025 at 04:25:13 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 77

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 112

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

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

✓ Sympy. Time used: 0.361 (sec). Leaf size: 70

from sympy import * 
x = symbols("x") 
a = symbols("a") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-n**2*y(x) + x*Derivative(y(x), x) + (a**2 + 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)
 

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