55.33.1 problem 211

Internal problem ID [13984]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-7
Problem number : 211
Date solved : Thursday, October 02, 2025 at 09:08:22 AM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 31

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

\[ y = x \left (c_1 \sinh \left (\frac {\sqrt {-a}}{x}\right )+c_2 \cosh \left (\frac {\sqrt {-a}}{x}\right )\right ) \]

✓ Mathematica. Time used: 0.071 (sec). Leaf size: 52

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

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

✓ Sympy. Time used: 0.050 (sec). Leaf size: 58

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

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