2.2.59 Problem 62

Internal problem ID [13265]
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 : 62
Date solved : Wednesday, December 31, 2025 at 12:41:38 PM
CAS classification : [_rational, _Riccati]

2.2.59.1 Solved using first_order_ode_riccati

53.823 (sec)

Entering first order ode riccati solver

\begin{align*} \left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime }&=y^{2}+\left (a_{1} x +b_{1} \right ) y+a_{0} x^{2}+b_{0} x +c_{0} \\ \end{align*}

In canonical form the ODE is

\begin{align*} y' &= F(x,y)\\ &= \frac {a_{0} x^{2}+y a_{1} x +b_{0} x +y b_{1} +y^{2}+c_{0}}{a_{2} x^{2}+b_{2} x +c_{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 \(f_0(x)=\frac {a_{0} x^{2}}{a_{2} x^{2}+b_{2} x +c_{2}}+\frac {b_{0} x}{a_{2} x^{2}+b_{2} x +c_{2}}+\frac {c_{0}}{a_{2} x^{2}+b_{2} x +c_{2}}\), \(f_1(x)=\frac {a_{1} x}{a_{2} x^{2}+b_{2} x +c_{2}}+\frac {b_{1}}{a_{2} x^{2}+b_{2} x +c_{2}}\) and \(f_2(x)=\frac {1}{a_{2} x^{2}+b_{2} x +c_{2}}\). Let

\begin{align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {u}{a_{2} x^{2}+b_{2} x +c_{2}}} \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 a_{2} x +b_{2}}{\left (a_{2} x^{2}+b_{2} x +c_{2} \right )^{2}}\\ f_1 f_2 &=\frac {\frac {a_{1} x}{a_{2} x^{2}+b_{2} x +c_{2}}+\frac {b_{1}}{a_{2} x^{2}+b_{2} x +c_{2}}}{a_{2} x^{2}+b_{2} x +c_{2}}\\ f_2^2 f_0 &=\frac {\frac {a_{0} x^{2}}{a_{2} x^{2}+b_{2} x +c_{2}}+\frac {b_{0} x}{a_{2} x^{2}+b_{2} x +c_{2}}+\frac {c_{0}}{a_{2} x^{2}+b_{2} x +c_{2}}}{\left (a_{2} x^{2}+b_{2} x +c_{2} \right )^{2}} \end{align*}

Substituting the above terms back in equation (2) gives

\[ \frac {u^{\prime \prime }\left (x \right )}{a_{2} x^{2}+b_{2} x +c_{2}}-\left (-\frac {2 a_{2} x +b_{2}}{\left (a_{2} x^{2}+b_{2} x +c_{2} \right )^{2}}+\frac {\frac {a_{1} x}{a_{2} x^{2}+b_{2} x +c_{2}}+\frac {b_{1}}{a_{2} x^{2}+b_{2} x +c_{2}}}{a_{2} x^{2}+b_{2} x +c_{2}}\right ) u^{\prime }\left (x \right )+\frac {\left (\frac {a_{0} x^{2}}{a_{2} x^{2}+b_{2} x +c_{2}}+\frac {b_{0} x}{a_{2} x^{2}+b_{2} x +c_{2}}+\frac {c_{0}}{a_{2} x^{2}+b_{2} x +c_{2}}\right ) u \left (x \right )}{\left (a_{2} x^{2}+b_{2} x +c_{2} \right )^{2}} = 0 \]

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

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

\begin{equation} \tag{3} \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'}{\frac {u}{a_{2} x^{2}+b_{2} x +c_{2}}} \\ y &= \text {Expression too large to display} \\ \end{align*}

Doing change of constants, the above solution becomes

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

Summary of solutions found

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

2.2.59.2 ✓ Maple. Time used: 0.009 (sec). Leaf size: 404913

ode:=(a__2*x^2+b__2*x+c__2)*diff(y(x),x) = y(x)^2+(a__1*x+b__1)*y(x)+a__0*x^2+b__0*x+c__0; 
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) = (a__1*x-2*a__2*x+ 
b__1-b__2)/(a__2*x^2+b__2*x+c__2)*diff(y(x),x)-(a__0*x^2+b__0*x+c__0)/(a__2*x^2 
+b__2*x+c__2)^2*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 
         -> 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 2F1 ODE 
         <- hypergeometric successful 
      <- special function solution successful 
   <- Riccati to 2nd Order successful
 

Maple step by step

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

2.2.59.3 ✗ Mathematica

ode=(a2*x^2+b2*x+c2)*D[y[x],x]==y[x]^2+(a1*x+b1)*y[x]+a0*x^2+b0*x+c0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.2.59.4 ✗ Sympy

from sympy import * 
x = symbols("x") 
a__0 = symbols("a__0") 
a__1 = symbols("a__1") 
a__2 = symbols("a__2") 
b__0 = symbols("b__0") 
b__1 = symbols("b__1") 
b__2 = symbols("b__2") 
c__0 = symbols("c__0") 
c__2 = symbols("c__2") 
y = Function("y") 
ode = Eq(-a__0*x**2 - b__0*x - c__0 - (a__1*x + b__1)*y(x) + (a__2*x**2 + b__2*x + c__2)*Derivative(y(x), x) - y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out