2.2.24 Problem 26

Internal problem ID [13230]
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 : 26
Date solved : Sunday, January 18, 2026 at 06:49:22 PM
CAS classification : [_Riccati]

2.2.24.1 Solved using first_order_ode_riccati

1.059 (sec)

Entering first order ode riccati solver

\begin{align*} y^{\prime }&=y^{2}+a \,x^{n} y+b \,x^{n -1} \\ \end{align*}

In canonical form the ODE is

\begin{align*} y' &= F(x,y)\\ &= y^{2}+a \,x^{n} y+b \,x^{n -1} \end{align*}

This is a Riccati ODE. Comparing the ODE to solve

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

With Riccati ODE standard form

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

Shows that \(f_0(x)=\frac {b \,x^{n}}{x}\), \(f_1(x)=x^{n} a\) and \(f_2(x)=1\). Let

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

Substituting the above terms back in equation (2) gives

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

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 x \operatorname {KummerM}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )+c_2 x \operatorname {KummerU}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n +1}}{n +1}\right ) \end{equation}

Taking derivative gives

\begin{equation} \tag{4} u^{\prime }\left (x \right ) = c_1 \operatorname {KummerM}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )+c_1 \left (\left (\frac {a \,x^{n +1}}{n +1}+\frac {a -b}{a \left (n +1\right )}-\frac {n +2}{n +1}\right ) \operatorname {KummerM}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )+\left (\frac {n +2}{n +1}-\frac {a -b}{a \left (n +1\right )}\right ) \operatorname {KummerM}\left (-1+\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )\right ) \left (n +1\right )+c_2 \operatorname {KummerU}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )+c_2 \left (\left (\frac {a \,x^{n +1}}{n +1}+\frac {a -b}{a \left (n +1\right )}-\frac {n +2}{n +1}\right ) \operatorname {KummerU}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )-\operatorname {KummerU}\left (-1+\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )\right ) \left (n +1\right ) \end{equation}

Substituting equations (3,4) into (1) results in

\begin{align*} y &= \frac {-u'}{f_2 u} \\ y &= \frac {-u'}{u} \\ y &= -\frac {c_1 \operatorname {KummerM}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )+c_1 \left (\left (\frac {a \,x^{n +1}}{n +1}+\frac {a -b}{a \left (n +1\right )}-\frac {n +2}{n +1}\right ) \operatorname {KummerM}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )+\left (\frac {n +2}{n +1}-\frac {a -b}{a \left (n +1\right )}\right ) \operatorname {KummerM}\left (-1+\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )\right ) \left (n +1\right )+c_2 \operatorname {KummerU}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )+c_2 \left (\left (\frac {a \,x^{n +1}}{n +1}+\frac {a -b}{a \left (n +1\right )}-\frac {n +2}{n +1}\right ) \operatorname {KummerU}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )-\operatorname {KummerU}\left (-1+\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )\right ) \left (n +1\right )}{c_1 x \operatorname {KummerM}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )+c_2 x \operatorname {KummerU}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )} \\ \end{align*}

Doing change of constants, the above solution becomes

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

Simplifying the above gives

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

Summary of solutions found

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

2.2.24.2 ✓ Maple. Time used: 0.002 (sec). Leaf size: 267

ode:=diff(y(x),x) = y(x)^2+a*x^n*y(x)+b*x^(n-1); 
dsolve(ode,y(x), singsol=all);
 

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

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 sub-methods: 
   trying Riccati_symmetries 
   trying Riccati to 2nd Order 
   -> Calling odsolve with the ODE, diff(diff(y(x),x),x) = a*x^n*diff(y(x),x)-b 
*x^(n-1)*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 
      -> 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 
         -> elliptic 
         -> Legendre 
         -> Kummer 
            -> hyper3: Equivalence to 1F1 under a power @ Moebius 
            <- hyper3 successful: received ODE is equivalent to the 1F1 ODE 
         <- Kummer successful 
      <- special function solution successful 
   <- Riccati to 2nd Order successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=y \left (x \right )^{2}+a \,x^{13230} y \left (x \right )+b \,x^{13229} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=y \left (x \right )^{2}+a \,x^{13230} y \left (x \right )+b \,x^{13229} \end {array} \]

2.2.24.3 ✓ Mathematica. Time used: 0.297 (sec). Leaf size: 453

ode=D[y[x],x]==y[x]^2+a*x^n*y[x]+b*x^(n-1); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\left (x^n\right )^{\frac {1}{n}} \left (-(-1)^{\frac {1}{n+1}} n (n+2) a^{\frac {1}{n+1}} \operatorname {Hypergeometric1F1}\left (\frac {a-b}{n a+a},\frac {n+2}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )+x^n \left (-(-1)^{\frac {1}{n+1}} n (a-b) a^{\frac {1}{n+1}} \left (x^n\right )^{\frac {1}{n}} \operatorname {Hypergeometric1F1}\left (\frac {n a+2 a-b}{n a+a},\frac {2 n+3}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )+b c_1 \left (\frac {1}{n}+1\right )^{\frac {1}{n+1}} n^{\frac {1}{n+1}} (n+2) \operatorname {Hypergeometric1F1}\left (\frac {n a+a-b}{n a+a},\frac {2 n+1}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )\right )\right )}{n (n+2) x \left ((-1)^{\frac {1}{n+1}} a^{\frac {1}{n+1}} \left (x^n\right )^{\frac {1}{n}} \operatorname {Hypergeometric1F1}\left (\frac {a-b}{n a+a},\frac {n+2}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )+c_1 \left (\frac {1}{n}+1\right )^{\frac {1}{n+1}} n^{\frac {1}{n+1}} \operatorname {Hypergeometric1F1}\left (-\frac {b}{n a+a},\frac {n}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )\right )}\\ y(x)&\to \frac {b x^{n-1} \left (x^n\right )^{\frac {1}{n}} \operatorname {Hypergeometric1F1}\left (\frac {n a+a-b}{n a+a},\frac {2 n+1}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )}{n \operatorname {Hypergeometric1F1}\left (-\frac {b}{n a+a},\frac {n}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )} \end{align*}

2.2.24.4 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-a*x**n*y(x) - b*x**(n - 1) - y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -a*x**n*y(x) - b*x**(n - 1) - y(x)**2 + Derivative(y(x), x) cann
 

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')