2.34.1 Problem 238

Internal problem ID [13899]
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, January 01, 2026 at 03:57:53 AM
CAS classification : [[_Emden, _Fowler]]

2.34.1.1 second order bessel ode

0.292 (sec)

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

Entering second order bessel ode solverWriting the ode as

\begin{align*} x^{2} y^{\prime \prime }-y^{\prime } x +\frac {a y}{x^{4}} = 0\tag {1} \end{align*}

Bessel ode has the form

\begin{align*} x^{2} y^{\prime \prime }+y^{\prime } x +\left (-n^{2}+x^{2}\right ) y = 0\tag {2} \end{align*}

The generalized form of Bessel ode is given by Bowman (1958) as the following

\begin{align*} x^{2} y^{\prime \prime }+\left (1-2 \alpha \right ) x y^{\prime }+\left (\beta ^{2} \gamma ^{2} x^{2 \gamma }-n^{2} \gamma ^{2}+\alpha ^{2}\right ) y = 0\tag {3} \end{align*}

With the standard solution

\begin{align*} y&=x^{\alpha } \left (c_1 \operatorname {BesselJ}\left (n , \beta \,x^{\gamma }\right )+c_2 \operatorname {BesselY}\left (n , \beta \,x^{\gamma }\right )\right )\tag {4} \end{align*}

Comparing (3) to (1) and solving for \(\alpha ,\beta ,n,\gamma \) gives

\begin{align*} \alpha &= 1\\ \beta &= \frac {\sqrt {a}}{2}\\ n &= {\frac {1}{2}}\\ \gamma &= -2 \end{align*}

Substituting all the above into (4) gives the solution as

\begin{align*} y = \frac {2 c_1 x \sin \left (\frac {\sqrt {a}}{2 x^{2}}\right )}{\sqrt {\pi }\, \sqrt {\frac {\sqrt {a}}{x^{2}}}}-\frac {2 c_2 x \cos \left (\frac {\sqrt {a}}{2 x^{2}}\right )}{\sqrt {\pi }\, \sqrt {\frac {\sqrt {a}}{x^{2}}}} \end{align*}

Summary of solutions found

\begin{align*} y &= \frac {2 c_1 x \sin \left (\frac {\sqrt {a}}{2 x^{2}}\right )}{\sqrt {\pi }\, \sqrt {\frac {\sqrt {a}}{x^{2}}}}-\frac {2 c_2 x \cos \left (\frac {\sqrt {a}}{2 x^{2}}\right )}{\sqrt {\pi }\, \sqrt {\frac {\sqrt {a}}{x^{2}}}} \\ \end{align*}

2.34.1.2 ✓ Maple. Time used: 0.004 (sec). Leaf size: 35

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

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

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
checking if the LODE is of Euler type 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Trying a Liouvillian solution using Kovacics algorithm 
   A Liouvillian solution exists 
   Group is reducible or imprimitive 
<- Kovacics algorithm successful
 

2.34.1.3 ✓ 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*}

2.34.1.4 ✓ 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)
 

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