2.2.21 Problem 23

Internal problem ID [13227]
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 : 23
Date solved : Wednesday, December 31, 2025 at 12:15:32 PM
CAS classification : [_rational, _Riccati]

2.2.21.1 Solved using first_order_ode_riccati

10.206 (sec)

Entering first order ode riccati solver

\begin{align*} \left (x^{n} a +b \,x^{m}+c \right ) \left (y^{\prime }-y^{2}\right )+a n \left (n -1\right ) x^{n -2}+b m \left (m -1\right ) x^{m -2}&=0 \\ \end{align*}

In canonical form the ODE is

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

This is a Riccati ODE. Comparing the ODE to solve

\[ y' = \textit {the\_rhs} \]

With Riccati ODE standard form

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

Shows that \(f_0(x)=-\frac {a \,n^{2} x^{n}}{\left (x^{n} a +b \,x^{m}+c \right ) x^{2}}-\frac {b \,m^{2} x^{m}}{\left (x^{n} a +b \,x^{m}+c \right ) x^{2}}+\frac {a \,x^{n} n}{\left (x^{n} a +b \,x^{m}+c \right ) x^{2}}+\frac {b m \,x^{m}}{\left (x^{n} a +b \,x^{m}+c \right ) x^{2}}\), \(f_1(x)=0\) 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 &=0\\ f_2^2 f_0 &=-\frac {a \,n^{2} x^{n}}{\left (x^{n} a +b \,x^{m}+c \right ) x^{2}}-\frac {b \,m^{2} x^{m}}{\left (x^{n} a +b \,x^{m}+c \right ) x^{2}}+\frac {a \,x^{n} n}{\left (x^{n} a +b \,x^{m}+c \right ) x^{2}}+\frac {b m \,x^{m}}{\left (x^{n} a +b \,x^{m}+c \right ) x^{2}} \end{align*}

Substituting the above terms back in equation (2) gives

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

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 \left (x^{n} a +b \,x^{m}+c \right )+c_2 \left (x^{n} a +b \,x^{m}+c \right ) \int \frac {1}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}d x \end{equation}

Taking derivative gives

\begin{equation} \tag{4} u^{\prime }\left (x \right ) = c_1 \left (\frac {a n \,x^{n}}{x}+\frac {b \,x^{m} m}{x}\right )+c_2 \left (\frac {a n \,x^{n}}{x}+\frac {b \,x^{m} m}{x}\right ) \int \frac {1}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}d x +\frac {c_2}{x^{n} a +b \,x^{m}+c} \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 \left (\frac {a n \,x^{n}}{x}+\frac {b \,x^{m} m}{x}\right )+c_2 \left (\frac {a n \,x^{n}}{x}+\frac {b \,x^{m} m}{x}\right ) \int \frac {1}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}d x +\frac {c_2}{x^{n} a +b \,x^{m}+c}}{c_1 \left (x^{n} a +b \,x^{m}+c \right )+c_2 \left (x^{n} a +b \,x^{m}+c \right ) \int \frac {1}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}d x} \\ \end{align*}

Doing change of constants, the above solution becomes

\[ y = -\frac {\frac {a n \,x^{n}}{x}+\frac {b \,x^{m} m}{x}+c_3 \left (\frac {a n \,x^{n}}{x}+\frac {b \,x^{m} m}{x}\right ) \int \frac {1}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}d x +\frac {c_3}{x^{n} a +b \,x^{m}+c}}{x^{n} a +b \,x^{m}+c +c_3 \left (x^{n} a +b \,x^{m}+c \right ) \int \frac {1}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}d x} \]

Summary of solutions found

\begin{align*} y &= -\frac {\frac {a n \,x^{n}}{x}+\frac {b \,x^{m} m}{x}+c_3 \left (\frac {a n \,x^{n}}{x}+\frac {b \,x^{m} m}{x}\right ) \int \frac {1}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}d x +\frac {c_3}{x^{n} a +b \,x^{m}+c}}{x^{n} a +b \,x^{m}+c +c_3 \left (x^{n} a +b \,x^{m}+c \right ) \int \frac {1}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}d x} \\ \end{align*}

2.2.21.2 ✓ Maple. Time used: 0.064 (sec). Leaf size: 169

ode:=(a*x^n+b*x^m+c)*(diff(y(x),x)-y(x)^2)+a*n*(n-1)*x^(-2+n)+b*m*(m-1)*x^(m-2) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

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*n^2*x^(n-2)+b*m^2 
*x^(m-2)-a*n*x^(n-2)-b*m*x^(m-2))/(a*x^n+b*x^m+c)*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)] 
      <- linear_1 successful 
   <- Riccati to 2nd Order successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (a \,x^{13227}+b \,x^{m}+c \right ) \left (\frac {d}{d x}y \left (x \right )-y \left (x \right )^{2}\right )[phone] a \,x^{13225}+b m \left (m -1\right ) x^{m -2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {a \,x^{13227} y \left (x \right )^{2}+b \,x^{m} y \left (x \right )^{2}+c y \left (x \right )^{2}-174940302 a \,x^{13225}-b \,m^{2} x^{m -2}+b m \,x^{m -2}}{a \,x^{13227}+b \,x^{m}+c} \end {array} \]

2.2.21.3 ✓ Mathematica. Time used: 3.419 (sec). Leaf size: 201

ode=(a*x^n+b*x^m+c)*(D[y[x],x]-y[x]^2)+a*n*(n-1)*x^(n-2)+b*m*(m-1)*x^(m-2)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {c_1 \left (\frac {\left (a n x^n+b m x^m\right ) \int _1^x\frac {1}{\left (b K[1]^m+a K[1]^n+c\right )^2}dK[1]}{x}+\frac {1}{a x^n+b x^m+c}\right )+a n x^{n-1}+b m x^{m-1}}{\left (a x^n+b x^m+c\right ) \left (1+c_1 \int _1^x\frac {1}{\left (b K[1]^m+a K[1]^n+c\right )^2}dK[1]\right )}\\ y(x)&\to -\frac {\frac {1}{\int _1^x\frac {1}{\left (b K[1]^m+a K[1]^n+c\right )^2}dK[1]}+\frac {\left (a n x^n+b m x^m\right ) \left (a x^n+b x^m+c\right )}{x}}{\left (a x^n+b x^m+c\right )^2} \end{align*}

2.2.21.4 ✗ 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*n*x**(n - 2)*(n - 1) + b*m*x**(m - 2)*(m - 1) + (-y(x)**2 + Derivative(y(x), x))*(a*x**n + b*x**m + c),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out