2.2.9 Problem 10

Internal problem ID [13215]
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 : 10
Date solved : Sunday, January 18, 2026 at 06:44:04 PM
CAS classification : [_Riccati]

2.2.9.1 Solved using first_order_ode_riccati_by_guessing_particular_solution

25.292 (sec)

Entering first order ode riccati guess solver

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

This is a Riccati ODE. Comparing the above ODE to solve with the Riccati standard form

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

Shows that

\begin{align*} f_0(x) & =\frac {b \,x^{m} m}{x}-a \,b^{2} x^{n} x^{2 m}\\ f_1(x) & =0\\ f_2(x) &=x^{n} a \end{align*}

Using trial and error, the following particular solution was found

\[ y_p = b \,x^{m} \]

Since a particular solution is known, then the general solution is given by

\begin{align*} y &= y_p + \frac { \phi (x) }{ c_1 - \int { \phi (x) f_2 \,dx} } \end{align*}

Where

\begin{align*} \phi (x) &= e^{ \int 2 f_2 y_p + f_1 \,dx } \end{align*}

Evaluating the above gives the general solution as

\[ y = b \,x^{m}+\frac {{\mathrm e}^{\int 2 x^{n} a b \,x^{m}d x}}{c_1 -\int {\mathrm e}^{\int 2 x^{n} a b \,x^{m}d x} x^{n} a d x} \]

Summary of solutions found

\begin{align*} y &= b \,x^{m}+\frac {{\mathrm e}^{\int 2 x^{n} a b \,x^{m}d x}}{c_1 -\int {\mathrm e}^{\int 2 x^{n} a b \,x^{m}d x} x^{n} a d x} \\ \end{align*}

2.2.9.2 ✓ Maple. Time used: 0.002 (sec). Leaf size: 512

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

\[ y = \frac {\left (-3 \left (\frac {\left (m +2 n +2\right ) \left (m +n +1\right ) x^{-\frac {3 m}{2}}}{3}+a \,x^{-\frac {m}{2}} b x \,x^{n} \left (m +\frac {4 n}{3}+\frac {4}{3}\right )\right ) c_1 \left (m +2 n +2\right ) \operatorname {WhittakerM}\left (\frac {m +2 n +2}{2 n +2 m +2}, \frac {2 m +3 n +3}{2 n +2 m +2}, -\frac {2 a b \,x^{m} x^{n} x}{m +n +1}\right )+2 \left (m +n +1\right ) \left (-\frac {\left (m +2 n +2\right ) \left (m +n +1\right ) x^{-\frac {3 m}{2}}}{2}+b \left (\left (-n -1-\frac {m}{2}\right ) x^{-\frac {m}{2}}+a \,x^{\frac {m}{2}} b x \,x^{n}\right ) a \,x^{n} x \right ) c_1 \operatorname {WhittakerM}\left (-\frac {m}{2 n +2 m +2}, \frac {2 m +3 n +3}{2 n +2 m +2}, -\frac {2 a b \,x^{m} x^{n} x}{m +n +1}\right )+2 x^{n} \left (\left (m +\frac {3 n}{2}+\frac {3}{2}\right ) x^{-\frac {3 m}{2}-n -1} c_1 \left (m +2 n +2\right )^{2} {\mathrm e}^{\frac {a b \,x^{m +n +1}}{m +n +1}} \left (-\frac {2 a b \,x^{m +n +1}}{m +n +1}\right )^{\frac {3 m +4 n +4}{2 n +2 m +2}}+a b \,x^{m +n +1} {\mathrm e}^{-\frac {a b \,x^{m +n +1}}{m +n +1}}\right ) x \right ) x^{-n}}{2 \left (-\frac {c_1 \,x^{-\frac {3 m}{2}} \left (m +2 n +2\right )^{2} \operatorname {WhittakerM}\left (\frac {m +2 n +2}{2 n +2 m +2}, \frac {2 m +3 n +3}{2 n +2 m +2}, -\frac {2 a b \,x^{m +n +1}}{m +n +1}\right )}{2}+\left (m +n +1\right ) c_1 \left (a b \,x^{n +1-\frac {m}{2}}-\frac {x^{-\frac {3 m}{2}} \left (m +2 n +2\right )}{2}\right ) \operatorname {WhittakerM}\left (-\frac {m}{2 n +2 m +2}, \frac {2 m +3 n +3}{2 n +2 m +2}, -\frac {2 a b \,x^{m +n +1}}{m +n +1}\right )+x^{n +1} {\mathrm e}^{-\frac {a b \,x^{m +n +1}}{m +n +1}}\right ) a x} \]

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) = n/x*diff(y(x),x)-a*x 
^n*b*(-a*b*x^(n+2*m)+m*x^(m-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 
            A Liouvillian solution exists 
            Reducible group (found an exponential solution) 
            Group is reducible, not completely reducible 
         <- Kovacics algorithm successful 
      <- Equivalence, under non-integer power transformations successful 
   <- Riccati to 2nd Order successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=a \,x^{13215} y \left (x \right )^{2}+b m \,x^{m -1}-a \,b^{2} x^{13215+2 m} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=a \,x^{13215} y \left (x \right )^{2}+b m \,x^{m -1}-a \,b^{2} x^{13215+2 m} \end {array} \]

2.2.9.3 ✓ Mathematica. Time used: 1.099 (sec). Leaf size: 306

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

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

2.2.9.4 ✗ Sympy

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

NotImplementedError : The given ODE a*b**2*x**(2*m + n) - a*x**n*y(x)**2 - b*m*x**(m - 1) + Derivati
 

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')