23.3.527 problem 533

Internal problem ID [6241]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 533
Date solved : Tuesday, September 30, 2025 at 02:38:50 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} a^{2} y+x^{4} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 23

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

\[ y = x \left (c_1 \sin \left (\frac {a}{x}\right )+c_2 \cos \left (\frac {a}{x}\right )\right ) \]

✓ Mathematica. Time used: 0.136 (sec). Leaf size: 42

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

\begin{align*} y(x)&\to c_1 x e^{\frac {i a}{x}}-\frac {i c_2 x e^{-\frac {i a}{x}}}{2 a} \end{align*}

✓ Sympy. Time used: 0.049 (sec). Leaf size: 44

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

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