2.2.63 Problem 66

Internal problem ID [13269]
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 : 66
Date solved : Friday, December 19, 2025 at 02:34:02 AM
CAS classification : [_rational, _Riccati]

\begin{align*} x^{3} y^{\prime }&=a \,x^{3} y^{2}+x \left (b x +c \right ) y+\alpha x +\beta \\ \end{align*}

Entering first order ode riccati solver

\begin{align*} x^{3} y^{\prime }&=a \,x^{3} y^{2}+x \left (b x +c \right ) y+\alpha x +\beta \\ \end{align*}

In canonical form the ODE is

\begin{align*} y' &= F(x,y)\\ &= \frac {a \,x^{3} y^{2}+b \,x^{2} y +c x y +\alpha x +\beta }{x^{3}} \end{align*}

This is a Riccati ODE. Comparing the ODE to solve

\[ y' = a \,y^{2}+\frac {b y}{x}+\frac {c y}{x^{2}}+\frac {\alpha }{x^{2}}+\frac {\beta }{x^{3}} \]

With Riccati ODE standard form

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

Shows that \(f_0(x)=\frac {\alpha x +\beta }{x^{3}}\), \(f_1(x)=\frac {b \,x^{2}+c x}{x^{3}}\) and \(f_2(x)=a\). Let

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

Substituting the above terms back in equation (2) gives

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

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 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}}{2}} {\mathrm e}^{-\frac {c}{x}} \operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )+c_2 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}}{2}} {\mathrm e}^{-\frac {c}{x}} \operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right ) \end{equation}

Taking derivative gives

\begin{equation} \tag{4} u^{\prime }\left (x \right ) = \frac {c_1 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}}{2}} \left (\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}}{2}\right ) {\mathrm e}^{-\frac {c}{x}} \operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )}{x}+\frac {c_1 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}}{2}} c \,{\mathrm e}^{-\frac {c}{x}} \operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )}{x^{2}}-\frac {c_1 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}}{2}} {\mathrm e}^{-\frac {c}{x}} \left (\operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}+1, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )-\operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )\right ) \left (\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta \right )}{2 c x}+\frac {c_2 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}}{2}} \left (\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}}{2}\right ) {\mathrm e}^{-\frac {c}{x}} \operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )}{x}+\frac {c_2 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}}{2}} c \,{\mathrm e}^{-\frac {c}{x}} \operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )}{x^{2}}-\frac {c_2 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}}{2}} {\mathrm e}^{-\frac {c}{x}} \left (\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}-\sqrt {-4 a \alpha +b^{2}+2 b +1}\right ) \operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}+1, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )-\operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )\right ) \left (\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta \right )}{2 c x} \end{equation}

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

\begin{align*} y &= \frac {-u'}{f_2 u} \\ y &= \frac {-u'}{a u} \\ y &= \frac {c_2 \left (\left (a \alpha +b +2\right ) c^{2}-\beta a \left (b +3\right ) c +a^{2} \beta ^{2}\right ) x \operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )+\left (\left (-c^{2}-c \left (b +2\right ) x +a x \beta \right ) c_1 \operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )+\left (-c^{2}-c \left (b +2\right ) x +a x \beta \right ) c_2 \operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )-\left (-\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c}{2}+\frac {\left (-b -3\right ) c}{2}+a \beta \right ) x \operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right ) c_1 \right ) c}{c^{2} x^{2} a \left (\operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right ) c_2 +\operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right ) c_1 \right )} \\ \end{align*}

Doing change of constants, the above solution becomes

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

Simplifying the above gives

\begin{align*} y &= \frac {\left (\left (a \alpha +b +2\right ) c^{2}-\beta a \left (b +3\right ) c +a^{2} \beta ^{2}\right ) x c_3 \operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )-c \left (\left (c^{2}+c \left (b +2\right ) x -a x \beta \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )-\left (-c^{2}-c \left (b +2\right ) x +a x \beta \right ) c_3 \operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )+\left (-\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c}{2}+\frac {\left (-b -3\right ) c}{2}+a \beta \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right ) x \right )}{c^{2} x^{2} a \left (\operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right ) c_3 +\operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )\right )} \\ \end{align*}

The solution

\[ y = \frac {\left (\left (a \alpha +b +2\right ) c^{2}-\beta a \left (b +3\right ) c +a^{2} \beta ^{2}\right ) x c_3 \operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )-c \left (\left (c^{2}+c \left (b +2\right ) x -a x \beta \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )-\left (-c^{2}-c \left (b +2\right ) x +a x \beta \right ) c_3 \operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )+\left (-\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c}{2}+\frac {\left (-b -3\right ) c}{2}+a \beta \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right ) x \right )}{c^{2} x^{2} a \left (\operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right ) c_3 +\operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )\right )} \]

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

2.2.63.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 438

ode:=x^3*diff(y(x),x) = x^3*a*y(x)^2+x*(b*x+c)*y(x)+alpha*x+beta; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (\left (a \alpha +b +2\right ) c^{2}-a \beta \left (b +3\right ) c +a^{2} \beta ^{2}\right ) x c_1 \operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )-c \left (\left (c^{2}+x \left (b +2\right ) c -a \beta x \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )-\left (-c^{2}-x \left (b +2\right ) c +a \beta x \right ) c_1 \operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )+\left (-\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c}{2}+\frac {\left (-b -3\right ) c}{2}+a \beta \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right ) x \right )}{x^{2} c^{2} a \left (\operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right ) c_1 +\operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )\right )} \]

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 to 2nd Order 
   -> Calling odsolve with the ODE, diff(diff(y(x),x),x) = (b*x+c)/x^2*diff(y(x 
),x)-a*(alpha*x+beta)/x^3*y(x), y(x) 
      *** Sublevel 2 *** 
      Methods for second order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      checking if the LODE has constant coefficients 
      checking if the LODE is of Euler type 
      trying a symmetry of the form [xi=0, eta=F(x)] 
      checking if the LODE is missing y 
      -> Trying a Liouvillian solution using Kovacics algorithm 
      <- No Liouvillian solutions exists 
      -> Trying a solution in terms of special functions: 
         -> Bessel 
         -> elliptic 
         -> Legendre 
         -> Kummer 
            -> hyper3: Equivalence to 1F1 under a power @ Moebius 
            <- hyper3 successful: received ODE is equivalent to the 1F1 ODE 
         <- Kummer successful 
      <- special function solution successful 
   <- Riccati to 2nd Order successful
 

Maple step by step

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

2.2.63.2 ✓ Mathematica. Time used: 0.971 (sec). Leaf size: 1168

ode=x^3*D[y[x],x]==a*x^3*y[x]^2+x*(b*x+c)*y[x]+\[Alpha]*x+\[Beta]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

2.2.63.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
Alpha = symbols("Alpha") 
BETA = symbols("BETA") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(-Alpha*x - BETA - a*x**3*y(x)**2 + x**3*Derivative(y(x), x) - x*(b*x + c)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0