23.3.538 problem 544

Internal problem ID [6252]
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 : 544
Date solved : Tuesday, September 30, 2025 at 02:38:58 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} a^{2} y+2 x^{3} y^{\prime }+x^{4} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 21

ode:=a^2*y(x)+2*x^3*diff(y(x),x)+x^4*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 25

ode=a^2*y[x] + 2*x^3*D[y[x],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 \cos \left (\frac {a}{x}\right )-c_2 \sin \left (\frac {a}{x}\right ) \end{align*}

✓ Sympy. Time used: 0.134 (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)) + 2*x**3*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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