55.34.1 problem 238

Internal problem ID [14012]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-8. Other equations.
Problem number : 238
Date solved : Thursday, October 02, 2025 at 09:08:53 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} a y-x^{5} y^{\prime }+x^{6} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 35

ode:=x^6*diff(diff(y(x),x),x)-x^5*diff(y(x),x)+a*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{2} \left (c_1 \sinh \left (\frac {\sqrt {-a}}{2 x^{2}}\right )+c_2 \cosh \left (\frac {\sqrt {-a}}{2 x^{2}}\right )\right ) \]

✓ Mathematica. Time used: 0.089 (sec). Leaf size: 58

ode=x^6*D[y[x],{x,2}]-x^5*D[y[x],x]+a*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} x^2 e^{-\frac {i \sqrt {a}}{2 x^2}} \left (2 c_1 e^{\frac {i \sqrt {a}}{x^2}}-\frac {i c_2}{\sqrt {a}}\right ) \end{align*}

✓ Sympy. Time used: 0.131 (sec). Leaf size: 65

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*y(x) + x**6*Derivative(y(x), (x, 2)) - x**5*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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