2.8.12 Problem 21
Internal
problem
ID
[13364]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.5-2
Problem
number
:
21
Date
solved
:
Sunday, January 18, 2026 at 07:31:40 PM
CAS
classification
:
[_Riccati]
2.8.12.1 Solved using first_order_ode_riccati
0.993 (sec)
Entering first order ode riccati solver
\begin{align*}
x^{2} y^{\prime }&=a^{2} x^{2} y^{2}-y x +b^{2} \ln \left (x \right )^{n} \\
\end{align*}
In canonical form the ODE is \begin{align*} y' &= F(x,y)\\ &= \frac {a^{2} x^{2} y^{2}-y x +b^{2} \ln \left (x \right )^{n}}{x^{2}} \end{align*}
This is a Riccati ODE. Comparing the ODE to solve
\[
y' = a^{2} y^{2}-\frac {y}{x}+\frac {b^{2} \ln \left (x \right )^{n}}{x^{2}}
\]
With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \]
Shows
that \(f_0(x)=\frac {b^{2} \ln \left (x \right )^{n}}{x^{2}}\), \(f_1(x)=-\frac {1}{x}\) and \(f_2(x)=a^{2}\). Let \begin{align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{u \,a^{2}} \tag {1} \end{align*}
Using the above substitution in the given ODE results (after some simplification) in a second
order ODE to solve for \(u(x)\) which is
\begin{align*} f_2 u''(x) -\left ( f_2' + f_1 f_2 \right ) u'(x) + f_2^2 f_0 u(x) &= 0 \tag {2} \end{align*}
But
\begin{align*} f_2' &=0\\ f_1 f_2 &=-\frac {a^{2}}{x}\\ f_2^2 f_0 &=\frac {a^{4} b^{2} \ln \left (x \right )^{n}}{x^{2}} \end{align*}
Substituting the above terms back in equation (2) gives
\[
a^{2} u^{\prime \prime }\left (x \right )+\frac {a^{2} u^{\prime }\left (x \right )}{x}+\frac {a^{4} b^{2} \ln \left (x \right )^{n} u \left (x \right )}{x^{2}} = 0
\]
Unable to solve. Will ask Maple to solve
this ode now.
The solution for \(u \left (x \right )\) is
\begin{equation}
\tag{3} u \left (x \right ) = c_1 \sqrt {\ln \left (x \right )}\, \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )+c_2 \sqrt {\ln \left (x \right )}\, \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )
\end{equation}
Taking derivative gives \begin{equation}
\tag{4} u^{\prime }\left (x \right ) = \frac {c_1 \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {\ln \left (x \right )}\, x}+\frac {2 c_1 \left (-\operatorname {BesselJ}\left (1+\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )+\frac {\ln \left (x \right )^{-1-\frac {n}{2}} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {a^{2} b^{2}}}\right ) \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}} \left (1+\frac {n}{2}\right )}{\sqrt {\ln \left (x \right )}\, x \left (n +2\right )}+\frac {c_2 \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {\ln \left (x \right )}\, x}+\frac {2 c_2 \left (-\operatorname {BesselY}\left (1+\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )+\frac {\ln \left (x \right )^{-1-\frac {n}{2}} \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {a^{2} b^{2}}}\right ) \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}} \left (1+\frac {n}{2}\right )}{\sqrt {\ln \left (x \right )}\, x \left (n +2\right )}
\end{equation}
Substituting equations (3,4) into (1) results in \begin{align*}
y &= \frac {-u'}{f_2 u} \\
y &= \frac {-u'}{u \,a^{2}} \\
y &= -\frac {\frac {c_1 \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {\ln \left (x \right )}\, x}+\frac {2 c_1 \left (-\operatorname {BesselJ}\left (1+\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )+\frac {\ln \left (x \right )^{-1-\frac {n}{2}} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {a^{2} b^{2}}}\right ) \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}} \left (1+\frac {n}{2}\right )}{\sqrt {\ln \left (x \right )}\, x \left (n +2\right )}+\frac {c_2 \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {\ln \left (x \right )}\, x}+\frac {2 c_2 \left (-\operatorname {BesselY}\left (1+\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )+\frac {\ln \left (x \right )^{-1-\frac {n}{2}} \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {a^{2} b^{2}}}\right ) \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}} \left (1+\frac {n}{2}\right )}{\sqrt {\ln \left (x \right )}\, x \left (n +2\right )}}{a^{2} \left (c_1 \sqrt {\ln \left (x \right )}\, \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )+c_2 \sqrt {\ln \left (x \right )}\, \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )\right )} \\
\end{align*}
Doing change of constants, the above solution becomes \[
y = -\frac {\frac {\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {\ln \left (x \right )}\, x}+\frac {2 \left (-\operatorname {BesselJ}\left (1+\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )+\frac {\ln \left (x \right )^{-1-\frac {n}{2}} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {a^{2} b^{2}}}\right ) \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}} \left (1+\frac {n}{2}\right )}{\sqrt {\ln \left (x \right )}\, x \left (n +2\right )}+\frac {c_3 \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {\ln \left (x \right )}\, x}+\frac {2 c_3 \left (-\operatorname {BesselY}\left (1+\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )+\frac {\ln \left (x \right )^{-1-\frac {n}{2}} \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {a^{2} b^{2}}}\right ) \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}} \left (1+\frac {n}{2}\right )}{\sqrt {\ln \left (x \right )}\, x \left (n +2\right )}}{a^{2} \left (\sqrt {\ln \left (x \right )}\, \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )+c_3 \sqrt {\ln \left (x \right )}\, \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )\right )}
\]
Simplifying the above gives
\begin{align*}
y &= \frac {\ln \left (x \right )^{1+\frac {n}{2}} \sqrt {a^{2} b^{2}}\, \operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right ) c_3 +\sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}} \operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )-c_3 \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )-\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )}{x \ln \left (x \right ) a^{2} \left (c_3 \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )+\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )\right )} \\
\end{align*}
Summary of solutions found
\begin{align*}
y &= \frac {\ln \left (x \right )^{1+\frac {n}{2}} \sqrt {a^{2} b^{2}}\, \operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right ) c_3 +\sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}} \operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )-c_3 \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )-\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )}{x \ln \left (x \right ) a^{2} \left (c_3 \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )+\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{1+\frac {n}{2}}}{n +2}\right )\right )} \\
\end{align*}
2.8.12.2 ✓ Maple. Time used: 0.003 (sec). Leaf size: 251
ode:=x^2*diff(y(x),x) = y(x)^2*a^2*x^2-y(x)*x+b^2*ln(x)^n;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\ln \left (x \right )^{\frac {n}{2}+1} \operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {a^{2} b^{2}}\, c_1 +\operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}-\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right ) c_1 -\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right )}{\left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right ) c_1 +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a^{2} b^{2}}\, \ln \left (x \right )^{\frac {n}{2}+1}}{n +2}\right )\right ) a^{2} x \ln \left (x \right )}
\]
Maple trace
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
differential order: 1; looking for linear symmetries
trying exact
Looking for potential symmetries
trying Riccati
trying Riccati sub-methods:
trying Riccati_symmetries
trying Riccati to 2nd Order
-> Calling odsolve with the ODE, diff(diff(y(x),x),x) = -1/x*diff(y(x),x)-a^
2/x^2*b^2*ln(x)^n*y(x), y(x)
*** Sublevel 2 ***
Methods for second order ODEs:
--- Trying classification methods ---
trying a symmetry of the form [xi=0, eta=F(x)]
checking if the LODE is missing y
-> Heun: Equivalence to the GHE or one of its 4 confluent cases under a \
power @ Moebius
-> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int\
(r(x), dx)) * 2F1([a1, a2], [b1], f)
-> Trying changes of variables to rationalize or make the ODE simpler
trying a symmetry of the form [xi=0, eta=F(x)]
checking if the LODE is missing y
-> Trying an equivalence, under non-integer power transformations,
to LODEs admitting Liouvillian solutions.
-> Trying a Liouvillian solution using Kovacics algorithm
<- No Liouvillian solutions exists
-> Trying a solution in terms of special functions:
-> Bessel
<- Bessel successful
<- special function solution successful
Change of variables used:
[x = exp(t)]
Linear ODE actually solved:
a^2*b^2*t^n*u(t)+diff(diff(u(t),t),t) = 0
<- change of variables successful
<- Riccati to 2nd Order successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left (\frac {d}{d x}y \left (x \right )\right )=a^{2} x^{2} y \left (x \right )^{2}-x y \left (x \right )+b^{2} \ln \left (x \right )^{13364} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {a^{2} x^{2} y \left (x \right )^{2}-x y \left (x \right )+b^{2} \ln \left (x \right )^{13364}}{x^{2}} \end {array} \]
2.8.12.3 ✓ Mathematica. Time used: 30.774 (sec). Leaf size: 2140
ode=x^2*D[y[x],x]==a^2*x^2*y[x]^2-x*y[x]+b^2*(Log[x])^n;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Too large to display
2.8.12.4 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
n = symbols("n")
y = Function("y")
ode = Eq(-a**2*x**2*y(x)**2 - b**2*log(x)**n + x**2*Derivative(y(x), x) + x*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -a**2*y(x)**2 - b**2*log(x)**n/x**2 + Derivative(y(x), x) + y(x)
Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0]
Sympy version 1.14.0
classify_ode(ode,func=y(x))
('factorable', 'lie_group')