69.24.4 problem 744

Internal problem ID [18503]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 18.2. Expanding a solution in generalized power series. Bessels equation. Exercises page 177
Problem number : 744
Date solved : Thursday, October 02, 2025 at 03:14:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (4 x^{2}-\frac {1}{9}\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 19

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

\[ y = c_1 \operatorname {BesselJ}\left (\frac {1}{3}, 2 x \right )+c_2 \operatorname {BesselY}\left (\frac {1}{3}, 2 x \right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 26

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+(4*x^2-1/9)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \operatorname {BesselJ}\left (\frac {1}{3},2 x\right )+c_2 \operatorname {BesselY}\left (\frac {1}{3},2 x\right ) \end{align*}

✓ Sympy. Time used: 0.139 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (4*x**2 - 1/9)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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