2.2.73 Problem 77

Internal problem ID [13279]
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 : 77
Date solved : Wednesday, December 31, 2025 at 12:52:57 PM
CAS classification : [_rational, _Riccati]

2.2.73.1 Solved using first_order_ode_riccati_by_guessing_particular_solution

3.850 (sec)

Entering first order ode riccati guess solver

\begin{align*} \left (x^{n} a +b \,x^{m}+c \right ) y^{\prime }&=\alpha \,x^{k} y^{2}+\beta \,x^{s} y-\alpha \,\lambda ^{2} x^{k}+\beta \lambda \,x^{s} \\ \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 {\alpha \,\lambda ^{2} x^{k}}{x^{n} a +b \,x^{m}+c}+\frac {\beta \lambda \,x^{s}}{x^{n} a +b \,x^{m}+c}\\ f_1(x) & =\frac {\beta \,x^{s}}{x^{n} a +b \,x^{m}+c}\\ f_2(x) &=\frac {x^{k} \alpha }{x^{n} a +b \,x^{m}+c} \end{align*}

Using trial and error, the following particular solution was found

\[ y_p = -\lambda \]

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

\[ y = -\lambda +\frac {{\mathrm e}^{\int \left (-\frac {2 \lambda \alpha \,x^{k}}{x^{n} a +b \,x^{m}+c}+\frac {\beta \,x^{s}}{x^{n} a +b \,x^{m}+c}\right )d x}}{c_1 -\int \frac {{\mathrm e}^{\int \left (-\frac {2 \lambda \alpha \,x^{k}}{x^{n} a +b \,x^{m}+c}+\frac {\beta \,x^{s}}{x^{n} a +b \,x^{m}+c}\right )d x} x^{k} \alpha }{x^{n} a +b \,x^{m}+c}d x} \]

Summary of solutions found

\begin{align*} y &= -\lambda +\frac {{\mathrm e}^{\int \left (-\frac {2 \lambda \alpha \,x^{k}}{x^{n} a +b \,x^{m}+c}+\frac {\beta \,x^{s}}{x^{n} a +b \,x^{m}+c}\right )d x}}{c_1 -\int \frac {{\mathrm e}^{\int \left (-\frac {2 \lambda \alpha \,x^{k}}{x^{n} a +b \,x^{m}+c}+\frac {\beta \,x^{s}}{x^{n} a +b \,x^{m}+c}\right )d x} x^{k} \alpha }{x^{n} a +b \,x^{m}+c}d x} \\ \end{align*}

2.2.73.2 ✓ Maple. Time used: 0.007 (sec). Leaf size: 164

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

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

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) = (x^n*a*k-a*x^n*n+x^m 
*b*k-b*x^m*m+beta*x^s*x+c*k)/(a*x^n+b*x^m+c)/x*diff(y(x),x)+x^k*alpha*lambda*(x 
^k*alpha*lambda-x^s*beta)/(a*x^n+b*x^m+c)^2*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 
         -> Whittaker 
            -> 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)-(1/(a*x^n+b*x^m+c)*x^k*alpha*y 
(x)^2+y(x)+1/(a*x^n+b*x^m+c)*beta*x^s*y(x)*x+x^2*(-1/(a*x^n+b*x^m+c)*alpha* 
lambda^2*x^k+1/(a*x^n+b*x^m+c)*beta*lambda*x^s))/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

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (a \,x^{13279}+b \,x^{m}+c \right ) \left (\frac {d}{d x}y \left (x \right )\right )=\alpha \,x^{k} y \left (x \right )^{2}+\beta \,x^{s} y \left (x \right )-\alpha \,\lambda ^{2} x^{k}+\beta \lambda \,x^{s} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\alpha \,x^{k} y \left (x \right )^{2}+\beta \,x^{s} y \left (x \right )-\alpha \,\lambda ^{2} x^{k}+\beta \lambda \,x^{s}}{a \,x^{13279}+b \,x^{m}+c} \end {array} \]

2.2.73.3 ✓ Mathematica. Time used: 3.721 (sec). Leaf size: 389

ode=(a*x^n+b*x^m+c)*D[y[x],x]==\[Alpha]*x^k*y[x]^2+\[Beta]*x^s*y[x]-\[Alpha]*\[Lambda]^2*x^k+\[Beta]*\[Lambda]*x^s; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[2]}-\frac {\beta K[1]^s-2 \alpha \lambda K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right ) \left (-\alpha \lambda K[2]^k+\alpha y(x) K[2]^k+\beta K[2]^s\right )}{(k-s) \alpha \beta \left (b K[2]^m+a K[2]^n+c\right ) (\lambda +y(x))}dK[2]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}-\frac {\beta K[1]^s-2 \alpha \lambda K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right ) K[2]^k}{(k-s) \beta \left (b K[2]^m+a K[2]^n+c\right ) (\lambda +K[3])}-\frac {\exp \left (-\int _1^{K[2]}-\frac {\beta K[1]^s-2 \alpha \lambda K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right ) \left (-\alpha \lambda K[2]^k+\alpha K[3] K[2]^k+\beta K[2]^s\right )}{(k-s) \alpha \beta \left (b K[2]^m+a K[2]^n+c\right ) (\lambda +K[3])^2}\right )dK[2]-\frac {\exp \left (-\int _1^x-\frac {\beta K[1]^s-2 \alpha \lambda K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right )}{(k-s) \alpha \beta (\lambda +K[3])^2}\right )dK[3]=c_1,y(x)\right ] \]

2.2.73.4 ✗ Sympy

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

Timed Out