2.2.27 Problem 30

Internal problem ID [13233]
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 : 30
Date solved : Wednesday, December 31, 2025 at 12:17:59 PM
CAS classification : [_Riccati]

2.2.27.1 Solved using first_order_ode_riccati_by_guessing_particular_solution

0.115 (sec)

Entering first order ode riccati guess solver

\begin{align*} y^{\prime }&=a \,x^{n} y^{2}+b \,x^{m} y+b \,x^{m} c -a \,c^{2} x^{n} \\ \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) & =-a \,c^{2} x^{n}+b \,x^{m} c\\ f_1(x) & =b \,x^{m}\\ f_2(x) &=x^{n} a \end{align*}

Using trial and error, the following particular solution was found

\[ y_p = -c \]

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

\[ y = -c +\frac {{\mathrm e}^{\frac {b \,x^{1+m}}{1+m}-\frac {2 a c \,x^{n +1}}{n +1}}}{c_1 -\int {\mathrm e}^{\frac {b \,x^{1+m}}{1+m}-\frac {2 a c \,x^{n +1}}{n +1}} x^{n} a d x} \]

Summary of solutions found

\begin{align*} y &= -c +\frac {{\mathrm e}^{\frac {b \,x^{1+m}}{1+m}-\frac {2 a c \,x^{n +1}}{n +1}}}{c_1 -\int {\mathrm e}^{\frac {b \,x^{1+m}}{1+m}-\frac {2 a c \,x^{n +1}}{n +1}} x^{n} a d x} \\ \end{align*}

2.2.27.2 ✓ Maple. Time used: 0.005 (sec). Leaf size: 88

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

\[ c_1 +a \int _{}^{x}\textit {\_a}^{n} {\mathrm e}^{-\frac {2 \left (-\frac {b \,\textit {\_a}^{m} \left (n +1\right )}{2}+a c \,\textit {\_a}^{n} \left (m +1\right )\right ) \textit {\_a}}{\left (m +1\right ) \left (n +1\right )}}d \textit {\_a} +\frac {{\mathrm e}^{-\frac {2 \left (-\frac {x^{m} b \left (n +1\right )}{2}+x^{n} a c \left (m +1\right )\right ) x}{\left (m +1\right ) \left (n +1\right )}}}{c +y} = 0 \]

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) = (x^m*b*x+n)/x*diff(y 
(x),x)+a*x^n*c*(x^n*a*c-x^m*b)*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 solution in terms of special functions: 
         -> Bessel 
         -> elliptic 
         -> Legendre 
         -> Kummer 
            -> hyper3: Equivalence to 1F1 under a power @ Moebius 
         -> hypergeometric 
            -> heuristic approach 
            -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius 
         -> Mathieu 
            -> Equivalence to the rational form of Mathieu ODE under a power @\ 
 Moebius 
      -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a \ 
power @ Moebius 
      -> 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 
      <- unable to find a useful change of variables 
         trying a symmetry of the form [xi=0, eta=F(x)] 
         trying 2nd order exact linear 
         trying symmetries linear in x and y(x) 
         trying to convert to a linear ODE with constant coefficients 
         trying 2nd order, integrating factor of the form mu(x,y) 
         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 solution in terms of special functions: 
            -> Bessel 
            -> elliptic 
            -> Legendre 
            -> Whittaker 
               -> hyper3: Equivalence to 1F1 under a power @ Moebius 
            -> hypergeometric 
               -> heuristic approach 
               -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebi\ 
us 
            -> Mathieu 
               -> Equivalence to the rational form of Mathieu ODE under a powe\ 
r @ Moebius 
         -> Heun: Equivalence to the GHE or one of its 4 confluent cases under \ 
a power @ Moebius 
         -> 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 
         <- unable to find a useful change of variables 
            trying a symmetry of the form [xi=0, eta=F(x)] 
         trying to convert to an ODE of Bessel type 
   -> Trying a change of variables to reduce to Bernoulli 
   -> Calling odsolve with the ODE, diff(y(x),x)-(a*x^n*y(x)^2+y(x)+x^m*b*y(x)* 
x+x^2*(b*c*x^m-a*c^2*x^n))/x, y(x), explicit 
      *** Sublevel 2 *** 
      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 inverse_Riccati 
      trying 1st order ODE linearizable_by_differentiation 
   -> trying a symmetry pattern of the form [F(x)*G(y), 0] 
   -> trying a symmetry pattern of the form [0, F(x)*G(y)] 
   <- symmetry pattern of the form [0, F(x)*G(y)] successful 
   <- Riccati with symmetry pattern of the form [0,F(x)*G(y)] successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=a \,x^{13233} y \left (x \right )^{2}+b \,x^{m} y \left (x \right )+b c \,x^{m}-a \,c^{2} x^{13233} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=a \,x^{13233} y \left (x \right )^{2}+b \,x^{m} y \left (x \right )+b c \,x^{m}-a \,c^{2} x^{13233} \end {array} \]

2.2.27.3 ✓ Mathematica. Time used: 0.951 (sec). Leaf size: 286

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

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {e^{\frac {b x^{m+1}}{m+1}-\frac {2 a c x^{n+1}}{n+1}}}{a b (m-n) (c+K[2])^2}-\int _1^x\left (-\frac {\exp \left (\frac {b K[1]^{m+1}}{m+1}-\frac {2 a c K[1]^{n+1}}{n+1}\right ) K[1]^n}{b (m-n) (c+K[2])}-\frac {\exp \left (\frac {b K[1]^{m+1}}{m+1}-\frac {2 a c K[1]^{n+1}}{n+1}\right ) \left (-b K[1]^m+a c K[1]^n-a K[2] K[1]^n\right )}{a b (m-n) (c+K[2])^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {\exp \left (\frac {b K[1]^{m+1}}{m+1}-\frac {2 a c K[1]^{n+1}}{n+1}\right ) \left (-b K[1]^m+a c K[1]^n-a y(x) K[1]^n\right )}{a b (m-n) (c+y(x))}dK[1]=c_1,y(x)\right ] \]

2.2.27.4 ✗ Sympy

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

NotImplementedError : The given ODE a*c**2*x**n - a*x**n*y(x)**2 - b*c*x**m - b*x**m*y(x) + Derivative(y(x), x) cannot be solved by the lie group method