23.3.533 problem 539

Internal problem ID [6247]
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 : 539
Date solved : Tuesday, September 30, 2025 at 02:38:54 PM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 19

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

\[ y = c_1 \operatorname {BesselJ}\left (0, \frac {1}{x}\right )+c_2 \operatorname {BesselY}\left (0, \frac {1}{x}\right ) \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 31

ode=y[x] + 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_2 \operatorname {BesselJ}\left (0,\frac {1}{x}\right )+\frac {c_1 K_0\left (\frac {i}{x}\right )}{\sqrt {\pi }} \end{align*}

✓ Sympy. Time used: 0.106 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 2)) + x**3*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} J_{0}\left (\frac {1}{x}\right ) + C_{2} Y_{0}\left (- \frac {1}{x}\right ) \]