2.2.4 Problem 4

Internal problem ID [13210]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number : 4
Date solved : Sunday, January 18, 2026 at 06:41:55 PM
CAS classification : [[_Riccati, _special]]

2.2.4.1 Solved using first_order_ode_reduced_riccati

0.141 (sec)

Entering first order ode reduced riccati solver

\begin{align*} y^{\prime }&=a y^{2}+b \,x^{n} \\ \end{align*}

This is reduced Riccati ode of the form

\begin{align*} y^{\prime }&=a \,x^{n}+b y^{2} \end{align*}

Comparing the given ode to the above shows that

\begin{align*} a &= b\\ b &= a\\ n &= n \end{align*}

Since \(n\neq -2\) then the solution of the reduced Riccati ode is given by

\begin{align*} w & =\sqrt {x}\left \{ \begin {array}[c]{cc} c_{1}\operatorname {BesselJ}\left ( \frac {1}{2k},\frac {1}{k}\sqrt {ab} x^{k}\right ) +c_{2}\operatorname {BesselY}\left ( \frac {1}{2k},\frac {1}{k}\sqrt {ab}x^{k}\right ) & ab>0\\ c_{1}\operatorname {BesselI}\left ( \frac {1}{2k},\frac {1}{k}\sqrt {-ab}x^{k}\right ) +c_{2}\operatorname {BesselK}\left ( \frac {1}{2k},\frac {1}{k}\sqrt {-ab}x^{k}\right ) & ab<0 \end {array} \right . \tag {1}\\ y & =-\frac {1}{b}\frac {w^{\prime }}{w}\nonumber \\ k &=1+\frac {n}{2}\nonumber \end{align*}

EQ(1) gives

\begin{align*} k &= 1+\frac {n}{2}\\ w &= \sqrt {x}\, \left (c_1 \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {\sqrt {a b}\, x^{1+\frac {n}{2}}}{1+\frac {n}{2}}\right )+c_2 \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {\sqrt {a b}\, x^{1+\frac {n}{2}}}{1+\frac {n}{2}}\right )\right ) \end{align*}

Therefore the solution becomes

\begin{align*} y & =-\frac {1}{b}\frac {w^{\prime }}{w} \end{align*}

Substituting the value of \(b,w\) found above and simplifying gives

\[ y = \frac {x^{1+\frac {n}{2}} \operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right ) \sqrt {a b}\, c_1 +\sqrt {a b}\, x^{1+\frac {n}{2}} \operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right ) c_2 -c_1 \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )-c_2 \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )}{x a \left (c_1 \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )+c_2 \operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )\right )} \]

Letting \(c_2 = 1\) the above becomes

\[ y = \frac {x^{1+\frac {n}{2}} \operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right ) \sqrt {a b}\, c_1 +\operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right ) \sqrt {a b}\, x^{1+\frac {n}{2}}-c_1 \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )-\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )}{x a \left (c_1 \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )+\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )\right )} \]

Summary of solutions found

\begin{align*} y &= \frac {x^{1+\frac {n}{2}} \operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right ) \sqrt {a b}\, c_1 +\operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right ) \sqrt {a b}\, x^{1+\frac {n}{2}}-c_1 \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )-\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )}{x a \left (c_1 \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )+\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )\right )} \\ \end{align*}

2.2.4.2 Solved using first_order_ode_riccati

0.237 (sec)

Entering first order ode riccati solver

\begin{align*} y^{\prime }&=a y^{2}+b \,x^{n} \\ \end{align*}

In canonical form the ODE is

\begin{align*} y' &= F(x,y)\\ &= a y^{2}+b \,x^{n} \end{align*}

This is a Riccati ODE. Comparing the ODE to solve

\[ y' = a y^{2}+b \,x^{n} \]

With Riccati ODE standard form

\[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \]

Shows that \(f_0(x)=b \,x^{n}\), \(f_1(x)=0\) and \(f_2(x)=a\). Let

\begin{align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{u a} \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 &=0\\ f_2^2 f_0 &=a^{2} b \,x^{n} \end{align*}

Substituting the above terms back in equation (2) gives

\[ a u^{\prime \prime }\left (x \right )+a^{2} b \,x^{n} u \left (x \right ) = 0 \]

Entering second order bessel ode solverWriting the ode as

\begin{align*} x^{2} \left (\frac {d^{2}u}{d x^{2}}\right )+b \,x^{n} a \,x^{2} u = 0\tag {1} \end{align*}

Bessel ode has the form

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

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

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

With the standard solution

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

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

\begin{align*} u = c_1 \sqrt {x}\, \operatorname {BesselJ}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )+c_2 \sqrt {x}\, \operatorname {BesselY}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right ) \end{align*}

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 b}\, x^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {x}}+\frac {2 c_1 \left (-\operatorname {BesselJ}\left (-\frac {1}{n +2}+1, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )-\frac {x^{-1-\frac {n}{2}} \operatorname {BesselJ}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {a b}}\right ) \sqrt {a b}\, x^{1+\frac {n}{2}} \left (1+\frac {n}{2}\right )}{\sqrt {x}\, \left (n +2\right )}+\frac {c_2 \operatorname {BesselY}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {x}}+\frac {2 c_2 \left (-\operatorname {BesselY}\left (-\frac {1}{n +2}+1, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )-\frac {x^{-1-\frac {n}{2}} \operatorname {BesselY}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {a b}}\right ) \sqrt {a b}\, x^{1+\frac {n}{2}} \left (1+\frac {n}{2}\right )}{\sqrt {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} \\ y &= -\frac {\frac {c_1 \operatorname {BesselJ}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {x}}+\frac {2 c_1 \left (-\operatorname {BesselJ}\left (-\frac {1}{n +2}+1, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )-\frac {x^{-1-\frac {n}{2}} \operatorname {BesselJ}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {a b}}\right ) \sqrt {a b}\, x^{1+\frac {n}{2}} \left (1+\frac {n}{2}\right )}{\sqrt {x}\, \left (n +2\right )}+\frac {c_2 \operatorname {BesselY}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {x}}+\frac {2 c_2 \left (-\operatorname {BesselY}\left (-\frac {1}{n +2}+1, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )-\frac {x^{-1-\frac {n}{2}} \operatorname {BesselY}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {a b}}\right ) \sqrt {a b}\, x^{1+\frac {n}{2}} \left (1+\frac {n}{2}\right )}{\sqrt {x}\, \left (n +2\right )}}{a \left (c_1 \sqrt {x}\, \operatorname {BesselJ}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )+c_2 \sqrt {x}\, \operatorname {BesselY}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )\right )} \\ \end{align*}

Doing change of constants, the above solution becomes

\[ y = -\frac {\frac {\operatorname {BesselJ}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {x}}+\frac {2 \left (-\operatorname {BesselJ}\left (-\frac {1}{n +2}+1, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )-\frac {x^{-1-\frac {n}{2}} \operatorname {BesselJ}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {a b}}\right ) \sqrt {a b}\, x^{1+\frac {n}{2}} \left (1+\frac {n}{2}\right )}{\sqrt {x}\, \left (n +2\right )}+\frac {c_3 \operatorname {BesselY}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {x}}+\frac {2 c_3 \left (-\operatorname {BesselY}\left (-\frac {1}{n +2}+1, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )-\frac {x^{-1-\frac {n}{2}} \operatorname {BesselY}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )}{2 \sqrt {a b}}\right ) \sqrt {a b}\, x^{1+\frac {n}{2}} \left (1+\frac {n}{2}\right )}{\sqrt {x}\, \left (n +2\right )}}{a \left (\sqrt {x}\, \operatorname {BesselJ}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )+c_3 \sqrt {x}\, \operatorname {BesselY}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )\right )} \]

Simplifying the above gives

\begin{align*} y &= \frac {x^{\frac {n}{2}} \sqrt {a b}\, \left (\operatorname {BesselY}\left (\frac {1+n}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right ) c_3 +\operatorname {BesselJ}\left (\frac {1+n}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )\right )}{a \left (c_3 \operatorname {BesselY}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )+\operatorname {BesselJ}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )\right )} \\ \end{align*}

Summary of solutions found

\begin{align*} y &= \frac {x^{\frac {n}{2}} \sqrt {a b}\, \left (\operatorname {BesselY}\left (\frac {1+n}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right ) c_3 +\operatorname {BesselJ}\left (\frac {1+n}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )\right )}{a \left (c_3 \operatorname {BesselY}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )+\operatorname {BesselJ}\left (-\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{n +2}\right )\right )} \\ \end{align*}

2.2.4.3 ✓ Maple. Time used: 0.003 (sec). Leaf size: 207

ode:=diff(y(x),x) = a*y(x)^2+b*x^n; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\operatorname {BesselJ}\left (\frac {3+n}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {a b}\, x^{\frac {n}{2}+1} c_1 +\operatorname {BesselY}\left (\frac {3+n}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {a b}\, x^{\frac {n}{2}+1}-c_1 \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right )-\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right )}{x a \left (c_1 \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right )+\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right )\right )} \]

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 Special 
<- Riccati Special successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=a y \left (x \right )^{2}+b \,x^{13210} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=a y \left (x \right )^{2}+b \,x^{13210} \end {array} \]

2.2.4.4 ✓ Mathematica. Time used: 0.255 (sec). Leaf size: 605

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

\begin{align*} y(x)&\to -\frac {\sqrt {a} \sqrt {b} x^{\frac {n}{2}+1} \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {1}{n+2}-1,\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )-\sqrt {a} \sqrt {b} x^{\frac {n}{2}+1} \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselJ}\left (1+\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )+\operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )-\sqrt {a} \sqrt {b} c_1 x^{\frac {n}{2}+1} \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )+\sqrt {a} \sqrt {b} c_1 x^{\frac {n}{2}+1} \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )+c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )}{2 a x \left (\operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )+c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )\right )}\\ y(x)&\to \frac {\frac {\sqrt {a} \sqrt {b} x^{n/2} \left (\operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )-\operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )\right )}{\operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )}-\frac {1}{x}}{2 a} \end{align*}

2.2.4.5 ✗ Sympy

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

NotImplementedError : The given ODE -a*y(x)**2 - b*x**n + Derivative(y(x), x) cannot be solved by th
 

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)) 
 
('1st_power_series', 'lie_group')