2.2.51 Problem 54

Internal problem ID [13257]
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 : 54
Date solved : Saturday, January 10, 2026 at 05:08:21 AM
CAS classification : [_rational, _Riccati]

\begin{align*} x^{2} y^{\prime }&=\left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y^{2}+\left (a \,x^{n}+b \right ) x y+c \,x^{2} \\ \end{align*}

Entering first order ode riccati solver

\begin{align*} x^{2} y^{\prime }&=\left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y^{2}+\left (a \,x^{n}+b \right ) x y+c \,x^{2} \\ \end{align*}

In canonical form the ODE is

\begin{align*} y' &= F(x,y)\\ &= \frac {x^{2 n} \alpha y^{2}+x^{n} \beta y^{2}+y x^{n} a x +\gamma y^{2}+b x y+c \,x^{2}}{x^{2}} \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 \(f_0(x)=c\), \(f_1(x)=\frac {x^{n} a}{x}+\frac {b}{x}\) and \(f_2(x)=\frac {\alpha \,x^{2 n}}{x^{2}}+\frac {x^{n} \beta }{x^{2}}+\frac {\gamma }{x^{2}}\). Let

\begin{align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{u \left (\frac {\alpha \,x^{2 n}}{x^{2}}+\frac {x^{n} \beta }{x^{2}}+\frac {\gamma }{x^{2}}\right )} \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' &=-\frac {2 \alpha \,x^{2 n}}{x^{3}}+\frac {2 \alpha \,x^{2 n} n}{x^{3}}-\frac {2 x^{n} \beta }{x^{3}}+\frac {x^{n} n \beta }{x^{3}}-\frac {2 \gamma }{x^{3}}\\ f_1 f_2 &=\left (\frac {x^{n} a}{x}+\frac {b}{x}\right ) \left (\frac {\alpha \,x^{2 n}}{x^{2}}+\frac {x^{n} \beta }{x^{2}}+\frac {\gamma }{x^{2}}\right )\\ f_2^2 f_0 &=\left (\frac {\alpha \,x^{2 n}}{x^{2}}+\frac {x^{n} \beta }{x^{2}}+\frac {\gamma }{x^{2}}\right )^{2} c \end{align*}

Substituting the above terms back in equation (2) gives

\[ \left (\frac {\alpha \,x^{2 n}}{x^{2}}+\frac {x^{n} \beta }{x^{2}}+\frac {\gamma }{x^{2}}\right ) u^{\prime \prime }\left (x \right )-\left (-\frac {2 \alpha \,x^{2 n}}{x^{3}}+\frac {2 \alpha \,x^{2 n} n}{x^{3}}-\frac {2 x^{n} \beta }{x^{3}}+\frac {x^{n} n \beta }{x^{3}}-\frac {2 \gamma }{x^{3}}+\left (\frac {x^{n} a}{x}+\frac {b}{x}\right ) \left (\frac {\alpha \,x^{2 n}}{x^{2}}+\frac {x^{n} \beta }{x^{2}}+\frac {\gamma }{x^{2}}\right )\right ) u^{\prime }\left (x \right )+\left (\frac {\alpha \,x^{2 n}}{x^{2}}+\frac {x^{n} \beta }{x^{2}}+\frac {\gamma }{x^{2}}\right )^{2} c u \left (x \right ) = 0 \]

Unable to solve. Will ask Maple to solve this ode now.

The solution for \(u \left (x \right )\) is

\begin{equation} \tag{3} \text {Expression too large to display} \end{equation}

Taking derivative gives

\begin{equation} \tag{4} \text {Expression too large to display} \end{equation}

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

\begin{align*} y &= \frac {-u'}{f_2 u} \\ y &= \frac {-u'}{u \left (\frac {\alpha \,x^{2 n}}{x^{2}}+\frac {x^{n} \beta }{x^{2}}+\frac {\gamma }{x^{2}}\right )} \\ y &= \text {Expression too large to display} \\ \end{align*}

Doing change of constants, the above solution becomes

\[ \text {Expression too large to display} \]

Simplifying the above gives

\begin{align*} \text {Expression too large to display} \\ \end{align*}

The above solution was found not to satisfy the ode or the IC. Hence it is removed.

2.2.51.1 ✓ Maple. Time used: 0.020 (sec). Leaf size: 215200

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

\[ \text {Expression too large to display} \]

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^(-2+n)*x^(-1+n)*a 
*beta*x^3+x^(-1+n)*x^(-2+2*n)*a*alpha*x^3+x^(-2+n)*b*beta*x^2+beta*x^(-2+n)*x^2 
*n+x^(-2+2*n)*alpha*b*x^2+2*alpha*x^(-2+2*n)*x^2*n-2*beta*x^(-2+n)*x^2+x^(-1+n) 
*a*gamma*x-2*alpha*x^(-2+2*n)*x^2+b*gamma-2*gamma)/x/(beta*x^(-2+n)*x^2+alpha*x 
^(-2+2*n)*x^2+gamma)*diff(y(x),x)-(beta*x^(-2+n)*x^2+alpha*x^(-2+2*n)*x^2+gamma 
)*c/x^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 Liouvillian solution using Kovacics algorithm 
         <- No Liouvillian solutions exists 
      -> 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 
                  <- hyper3 successful: received ODE is equivalent to the 1F1 O\ 
DE 
         <- hypergeometric successful 
      <- special function solution successful 
   <- Riccati to 2nd Order successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left (\frac {d}{d x}y \left (x \right )\right )=\left (\alpha \,x^{26514}+\beta \,x^{13257}+\gamma \right ) y \left (x \right )^{2}+\left (a \,x^{13257}+b \right ) x y \left (x \right )+c \,x^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\left (\alpha \,x^{26514}+\beta \,x^{13257}+\gamma \right ) y \left (x \right )^{2}+\left (a \,x^{13257}+b \right ) x y \left (x \right )+c \,x^{2}}{x^{2}} \end {array} \]

2.2.51.2 ✓ Mathematica. Time used: 1.909 (sec). Leaf size: 2649

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

Too large to display

2.2.51.3 ✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (c*x**2 + x*(a*x**n + b)*y(x) + (Alpha*x**(2*n) + BETA*x**n + Gamma)*y(x)**2)/x**2 cannot be solved by the factorable group method