2.2.25 Problem 27

Internal problem ID [13231]
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 : 27
Date solved : Wednesday, December 31, 2025 at 12:17:11 PM
CAS classification : [_Riccati]

2.2.25.1 Solved using first_order_ode_riccati

12.095 (sec)

Entering first order ode riccati solver

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

In canonical form the ODE is

\begin{align*} y' &= F(x,y)\\ &= a \,x^{2}+\alpha x y+b x +\beta y+y^{2}+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 \(f_0(x)=a \,x^{2}+b x +c\), \(f_1(x)=\alpha x +\beta \) 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 &=\alpha x +\beta \\ f_2^2 f_0 &=a \,x^{2}+b x +c \end{align*}

Substituting the above terms back in equation (2) gives

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

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 \operatorname {hypergeom}\left (\left [\frac {\left (\alpha ^{3}+2 \alpha ^{2} c +2 \left (-b \beta -2 a \right ) \alpha +2 \left (\beta ^{2}-4 c \right ) a +2 b^{2}\right ) \left (\alpha ^{2}-4 a \right )^{{3}/{2}}+64 \left (-\frac {\alpha ^{2}}{4}+a \right )^{3}}{4 \left (-\alpha ^{2}+4 a \right )^{3}}\right ], \left [\frac {1}{2}\right ], \frac {\left (-\alpha ^{2} x +4 a x -\alpha \beta +2 b \right )^{2}}{2 \left (\alpha ^{2}-4 a \right )^{{3}/{2}}}\right ) {\mathrm e}^{-\frac {4 x \left (\left (-\frac {\alpha x}{16}-\frac {\beta }{8}\right ) \left (\alpha ^{2}-4 a \right )^{{3}/{2}}+\left (-\frac {1}{4} \alpha ^{2} x +a x -\frac {1}{2} \alpha \beta +b \right ) \left (-\frac {\alpha ^{2}}{4}+a \right )\right )}{\left (\alpha ^{2}-4 a \right )^{{3}/{2}}}}+c_2 \operatorname {hypergeom}\left (\left [\frac {\left (\alpha ^{3}+2 \alpha ^{2} c +2 \left (-b \beta -2 a \right ) \alpha +2 \left (\beta ^{2}-4 c \right ) a +2 b^{2}\right ) \left (\alpha ^{2}-4 a \right )^{{3}/{2}}+192 \left (-\frac {\alpha ^{2}}{4}+a \right )^{3}}{4 \left (-\alpha ^{2}+4 a \right )^{3}}\right ], \left [\frac {3}{2}\right ], \frac {\left (-\alpha ^{2} x +4 a x -\alpha \beta +2 b \right )^{2}}{2 \left (\alpha ^{2}-4 a \right )^{{3}/{2}}}\right ) \left (-\alpha ^{2} x +4 a x -\alpha \beta +2 b \right ) {\mathrm e}^{-\frac {4 x \left (\left (-\frac {\alpha x}{16}-\frac {\beta }{8}\right ) \left (\alpha ^{2}-4 a \right )^{{3}/{2}}+\left (-\frac {1}{4} \alpha ^{2} x +a x -\frac {1}{2} \alpha \beta +b \right ) \left (-\frac {\alpha ^{2}}{4}+a \right )\right )}{\left (\alpha ^{2}-4 a \right )^{{3}/{2}}}} \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} \\ 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*}

Summary of solutions found

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

2.2.25.2 ✓ Maple. Time used: 0.003 (sec). Leaf size: 973

ode:=diff(y(x),x) = y(x)^2+(alpha*x+beta)*y(x)+a*x^2+b*x+c; 
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 to 2nd Order 
   -> Calling odsolve with the ODE, diff(diff(y(x),x),x) = (alpha*x+beta)*diff( 
y(x),x)+(-a*x^2-b*x-c)*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 
         -> hypergeometric 
            -> heuristic approach 
            -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius 
            <- hyper3 successful: indirect Equivalence to 0F1 under ``^ @ Moebi\ 
us`` is resolved 
         <- hypergeometric successful 
      <- special function solution successful 
   <- Riccati to 2nd Order successful
 

Maple step by step

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

2.2.25.3 ✓ Mathematica. Time used: 1.497 (sec). Leaf size: 1288

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

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

2.2.25.4 ✗ 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(-a*x**2 - b*x - c - (Alpha*x + BETA)*y(x) - y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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