34.6.8 problem 8

Internal problem ID [6082]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter IX, Special forms of differential equations. Examples XVII. page 247
Problem number : 8
Date solved : Wednesday, March 05, 2025 at 12:11:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-a^{2} y&=\frac {6 y}{x^{2}} \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 48

ode:=diff(diff(y(x),x),x)-a^2*y(x) = 6/x^2*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_2 \,{\mathrm e}^{-a x} \left (a^{2} x^{2}+3 a x +3\right )+c_1 \,{\mathrm e}^{a x} \left (a^{2} x^{2}-3 a x +3\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.126 (sec). Leaf size: 90

ode=D[y[x],{x,2}]-a^2*y[x]==6*y[x]/x^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {\sqrt {\frac {2}{\pi }} \left (\left (a^2 c_2 x^2-3 i a c_1 x+3 c_2\right ) \cosh (a x)+i \left (c_1 \left (a^2 x^2+3\right )+3 i a c_2 x\right ) \sinh (a x)\right )}{a^2 x^{3/2} \sqrt {-i a x}} \]

✓ Sympy. Time used: 0.100 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a**2*y(x) + Derivative(y(x), (x, 2)) - 6*y(x)/x**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \sqrt {x} \left (C_{1} J_{\frac {5}{2}}\left (x \sqrt {- a^{2}}\right ) + C_{2} Y_{\frac {5}{2}}\left (x \sqrt {- a^{2}}\right )\right ) \]