34.6.11 problem 11

Internal problem ID [6085]
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 : 11
Date solved : Sunday, March 30, 2025 at 10:38:09 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\frac {\left (-9 a^{2}+4 x^{2}\right ) y}{4 a^{2}}&=0 \end{align*}

✓ Maple. Time used: 0.051 (sec). Leaf size: 37

ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+1/4*(-9*a^2+4*x^2)/a^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.073 (sec). Leaf size: 62

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+(4*x^2-9*a^2)/(4*a^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\frac {\sqrt {\frac {2}{\pi }} \left ((a c_2+c_1 x) \cos \left (\frac {x}{a}\right )+(c_2 x-a c_1) \sin \left (\frac {x}{a}\right )\right )}{x \sqrt {\frac {x}{a}}} \]

✓ Sympy. Time used: 0.245 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (-9*a**2 + 4*x**2)*y(x)/(4*a**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} J_{\frac {3}{2}}\left (\frac {x}{a}\right ) + C_{2} Y_{\frac {3}{2}}\left (\frac {x}{a}\right ) \]