2.2.53 Problem 56

Internal problem ID [13259]
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 : 56
Date solved : Wednesday, December 31, 2025 at 12:28:42 PM
CAS classification : [_rational, _Riccati]

2.2.53.1 Solved using first_order_ode_riccati

544.721 (sec)

Entering first order ode riccati solver

\begin{align*} \left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\frac {b \left (a +\beta \right )}{\alpha }&=0 \\ \end{align*}

In canonical form the ODE is

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

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

Substituting the above terms back in equation (2) gives

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

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 (a \,x^{2}+b \right )^{-\frac {\beta }{4 a}} \operatorname {LegendreP}\left (\frac {\beta }{2 a}, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right )+c_2 \left (a \,x^{2}+b \right )^{-\frac {\beta }{4 a}} \operatorname {LegendreQ}\left (\frac {\beta }{2 a}, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right ) \end{equation}

Taking derivative gives

\begin{equation} \tag{4} u^{\prime }\left (x \right ) = -\frac {c_1 \left (a \,x^{2}+b \right )^{-\frac {\beta }{4 a}} \beta x \operatorname {LegendreP}\left (\frac {\beta }{2 a}, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right )}{2 \left (a \,x^{2}+b \right )}+\frac {c_1 \left (a \,x^{2}+b \right )^{-\frac {\beta }{4 a}} \left (\left (\frac {\beta }{2 a}-\frac {2 a +\beta }{2 a}+1\right ) \operatorname {LegendreP}\left (\frac {\beta }{2 a}+1, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right )-\frac {\left (\frac {\beta }{2 a}+1\right ) a x \operatorname {LegendreP}\left (\frac {\beta }{2 a}, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right )}{\sqrt {-a b}}\right ) a}{\sqrt {-a b}\, \left (-1-\frac {a \,x^{2}}{b}\right )}-\frac {c_2 \left (a \,x^{2}+b \right )^{-\frac {\beta }{4 a}} \beta x \operatorname {LegendreQ}\left (\frac {\beta }{2 a}, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right )}{2 \left (a \,x^{2}+b \right )}+\frac {c_2 \left (a \,x^{2}+b \right )^{-\frac {\beta }{4 a}} \left (\left (\frac {\beta }{2 a}-\frac {2 a +\beta }{2 a}+1\right ) \operatorname {LegendreQ}\left (\frac {\beta }{2 a}+1, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right )-\frac {\left (\frac {\beta }{2 a}+1\right ) a x \operatorname {LegendreQ}\left (\frac {\beta }{2 a}, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right )}{\sqrt {-a b}}\right ) a}{\sqrt {-a b}\, \left (-1-\frac {a \,x^{2}}{b}\right )} \end{equation}

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

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

Doing change of constants, the above solution becomes

\[ y = \frac {\left (-\frac {\left (a \,x^{2}+b \right )^{-\frac {\beta }{4 a}} \beta x \operatorname {LegendreP}\left (\frac {\beta }{2 a}, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right )}{2 \left (a \,x^{2}+b \right )}+\frac {\left (a \,x^{2}+b \right )^{-\frac {\beta }{4 a}} \left (\left (\frac {\beta }{2 a}-\frac {2 a +\beta }{2 a}+1\right ) \operatorname {LegendreP}\left (\frac {\beta }{2 a}+1, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right )-\frac {\left (\frac {\beta }{2 a}+1\right ) a x \operatorname {LegendreP}\left (\frac {\beta }{2 a}, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right )}{\sqrt {-a b}}\right ) a}{\sqrt {-a b}\, \left (-1-\frac {a \,x^{2}}{b}\right )}-\frac {c_3 \left (a \,x^{2}+b \right )^{-\frac {\beta }{4 a}} \beta x \operatorname {LegendreQ}\left (\frac {\beta }{2 a}, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right )}{2 \left (a \,x^{2}+b \right )}+\frac {c_3 \left (a \,x^{2}+b \right )^{-\frac {\beta }{4 a}} \left (\left (\frac {\beta }{2 a}-\frac {2 a +\beta }{2 a}+1\right ) \operatorname {LegendreQ}\left (\frac {\beta }{2 a}+1, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right )-\frac {\left (\frac {\beta }{2 a}+1\right ) a x \operatorname {LegendreQ}\left (\frac {\beta }{2 a}, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right )}{\sqrt {-a b}}\right ) a}{\sqrt {-a b}\, \left (-1-\frac {a \,x^{2}}{b}\right )}\right ) \left (a \,x^{2}+b \right )}{\alpha \left (\left (a \,x^{2}+b \right )^{-\frac {\beta }{4 a}} \operatorname {LegendreP}\left (\frac {\beta }{2 a}, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right )+c_3 \left (a \,x^{2}+b \right )^{-\frac {\beta }{4 a}} \operatorname {LegendreQ}\left (\frac {\beta }{2 a}, \frac {2 a +\beta }{2 a}, \frac {a x}{\sqrt {-a b}}\right )\right )} \]

Simplifying the above gives

\begin{align*} y &= -\frac {\left (a +\beta \right ) x}{\alpha } \\ \end{align*}

Summary of solutions found

\begin{align*} y &= -\frac {\left (a +\beta \right ) x}{\alpha } \\ \end{align*}

2.2.53.2 ✓ Maple. Time used: 0.013 (sec). Leaf size: 518

ode:=(a*x^2+b)*diff(y(x),x)+alpha*y(x)^2+beta*x*y(x)+b/alpha*(a+beta) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {a^{2} b \left (-\frac {\left (-\frac {-a x +\sqrt {-b a}}{2 \sqrt {-b a}}\right )^{\frac {\beta }{a}} \left (a \,x^{2}+b \right ) \left (a \,x^{2}+2 x \sqrt {-b a}-b \right ) \operatorname {HeunCPrime}\left (0, -1-\frac {\beta }{a}, 1+\frac {\beta }{2 a}, 0, \frac {1}{2}+\frac {\beta ^{2}}{4 a^{2}}+\frac {\beta }{2 a}, \frac {2 \sqrt {-b a}}{-a x +\sqrt {-b a}}\right )}{2}-2 \left (\left (3 a \,x^{2}-b \right ) \sqrt {-b a}+a x \left (a \,x^{2}-3 b \right )\right ) a c_1 b \operatorname {HeunCPrime}\left (0, \frac {\beta }{a}+1, 1+\frac {\beta }{2 a}, 0, \frac {1}{2}+\frac {\beta ^{2}}{4 a^{2}}+\frac {\beta }{2 a}, \frac {2 \sqrt {-b a}}{-a x +\sqrt {-b a}}\right )+\left (\frac {\left (-\frac {-a x +\sqrt {-b a}}{2 \sqrt {-b a}}\right )^{\frac {\beta }{a}} \left (a \,x^{2}-2 x \sqrt {-b a}-b \right ) \operatorname {hypergeom}\left (\left [1, -\frac {\beta }{2 a}\right ], \left [-\frac {\beta }{a}\right ], \frac {2 \sqrt {-b a}}{a x +\sqrt {-b a}}\right )}{4}+c_1 \left (\left (-a^{2} x^{2}+\left (-x^{2} \beta -2 b \right ) a -b \beta \right ) \sqrt {-b a}+a^{2} b x \right ) \left (\frac {a x -\sqrt {-b a}}{a x +\sqrt {-b a}}\right )^{\frac {\beta }{2 a}}\right ) \left (a \,x^{2}+b \right )\right )}{\alpha \left (\frac {\left (-\frac {-a x +\sqrt {-b a}}{2 \sqrt {-b a}}\right )^{\frac {\beta }{a}} \left (x \sqrt {-b a}+b \right ) \operatorname {hypergeom}\left (\left [1, -\frac {\beta }{2 a}\right ], \left [-\frac {\beta }{a}\right ], \frac {2 \sqrt {-b a}}{a x +\sqrt {-b a}}\right )}{4}+\sqrt {-b a}\, \left (\frac {a x -\sqrt {-b a}}{a x +\sqrt {-b a}}\right )^{\frac {\beta }{2 a}} c_1 a b \right ) \left (a x +\sqrt {-b a}\right ) \left (a x -\sqrt {-b a}\right )^{2}} \]

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: 
   <- Abel AIR successful: ODE belongs to the 2F1 3-parameter class
 

Maple step by step

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

2.2.53.3 ✓ Mathematica. Time used: 0.447 (sec). Leaf size: 27

ode=(a*x^2+b)*D[y[x],x]+\[Alpha]*y[x]^2+\[Beta]*x*y[x]+b/\[Alpha]*(a+\[Beta])==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {x (a+\beta )}{\alpha }\\ y(x)&\to -\frac {x (a+\beta )}{\alpha } \end{align*}

2.2.53.4 ✗ Sympy

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

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