2.2.17 Problem 19

Internal problem ID [13223]
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 : 19
Date solved : Wednesday, December 31, 2025 at 12:12:31 PM
CAS classification : [_rational, _Riccati]

2.2.17.1 Solved using first_order_ode_riccati

12.210 (sec)

Entering first order ode riccati solver

\begin{align*} a \,x^{2} \left (x -1\right )^{2} \left (y^{\prime }+\lambda y^{2}\right )+b \,x^{2}+c x +s&=0 \\ \end{align*}

In canonical form the ODE is

\begin{align*} y' &= F(x,y)\\ &= -\frac {y^{2} a \lambda \,x^{4}-2 y^{2} a \lambda \,x^{3}+a \,x^{2} \lambda y^{2}+b \,x^{2}+c x +s}{a \,x^{2} \left (x -1\right )^{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 \(f_0(x)=-\frac {b}{a \left (x -1\right )^{2}}-\frac {c}{a x \left (x -1\right )^{2}}-\frac {s}{a \,x^{2} \left (x -1\right )^{2}}\), \(f_1(x)=0\) and \(f_2(x)=-\frac {x^{2} \lambda }{\left (x -1\right )^{2}}+\frac {2 x \lambda }{\left (x -1\right )^{2}}-\frac {\lambda }{\left (x -1\right )^{2}}\). Let

\begin{align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{u \left (-\frac {x^{2} \lambda }{\left (x -1\right )^{2}}+\frac {2 x \lambda }{\left (x -1\right )^{2}}-\frac {\lambda }{\left (x -1\right )^{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 x^{2} \lambda }{\left (x -1\right )^{3}}-\frac {2 x \lambda }{\left (x -1\right )^{2}}-\frac {4 x \lambda }{\left (x -1\right )^{3}}+\frac {2 \lambda }{\left (x -1\right )^{2}}+\frac {2 \lambda }{\left (x -1\right )^{3}}\\ f_1 f_2 &=0\\ f_2^2 f_0 &=\left (-\frac {x^{2} \lambda }{\left (x -1\right )^{2}}+\frac {2 x \lambda }{\left (x -1\right )^{2}}-\frac {\lambda }{\left (x -1\right )^{2}}\right )^{2} \left (-\frac {b}{a \left (x -1\right )^{2}}-\frac {c}{a x \left (x -1\right )^{2}}-\frac {s}{a \,x^{2} \left (x -1\right )^{2}}\right ) \end{align*}

Substituting the above terms back in equation (2) gives

\[ \left (-\frac {x^{2} \lambda }{\left (x -1\right )^{2}}+\frac {2 x \lambda }{\left (x -1\right )^{2}}-\frac {\lambda }{\left (x -1\right )^{2}}\right ) u^{\prime \prime }\left (x \right )-\left (\frac {2 x^{2} \lambda }{\left (x -1\right )^{3}}-\frac {2 x \lambda }{\left (x -1\right )^{2}}-\frac {4 x \lambda }{\left (x -1\right )^{3}}+\frac {2 \lambda }{\left (x -1\right )^{2}}+\frac {2 \lambda }{\left (x -1\right )^{3}}\right ) u^{\prime }\left (x \right )+\left (-\frac {x^{2} \lambda }{\left (x -1\right )^{2}}+\frac {2 x \lambda }{\left (x -1\right )^{2}}-\frac {\lambda }{\left (x -1\right )^{2}}\right )^{2} \left (-\frac {b}{a \left (x -1\right )^{2}}-\frac {c}{a x \left (x -1\right )^{2}}-\frac {s}{a \,x^{2} \left (x -1\right )^{2}}\right ) u \left (x \right ) = 0 \]

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 \,x^{\frac {\sqrt {a}+\sqrt {-4 \lambda s +a}}{2 \sqrt {a}}} \left (x -1\right )^{\frac {\sqrt {a}-\sqrt {a -4 \lambda \left (b +c +s \right )}}{2 \sqrt {a}}} \operatorname {hypergeom}\left (\left [\frac {-\sqrt {a -4 \lambda \left (b +c +s \right )}+\sqrt {a}+\sqrt {-4 \lambda s +a}+\sqrt {-4 b \lambda +a}}{2 \sqrt {a}}, \frac {-\sqrt {a -4 \lambda \left (b +c +s \right )}+\sqrt {a}+\sqrt {-4 \lambda s +a}-\sqrt {-4 b \lambda +a}}{2 \sqrt {a}}\right ], \left [\frac {\sqrt {a}+\sqrt {-4 \lambda s +a}}{\sqrt {a}}\right ], x\right )+c_2 \,x^{\frac {\sqrt {a}-\sqrt {-4 \lambda s +a}}{2 \sqrt {a}}} \left (x -1\right )^{\frac {\sqrt {a}-\sqrt {a -4 \lambda \left (b +c +s \right )}}{2 \sqrt {a}}} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {a -4 \lambda \left (b +c +s \right )}-\sqrt {a}+\sqrt {-4 \lambda s +a}+\sqrt {-4 b \lambda +a}}{2 \sqrt {a}}, -\frac {\sqrt {a -4 \lambda \left (b +c +s \right )}-\sqrt {a}+\sqrt {-4 \lambda s +a}-\sqrt {-4 b \lambda +a}}{2 \sqrt {a}}\right ], \left [\frac {\sqrt {a}-\sqrt {-4 \lambda s +a}}{\sqrt {a}}\right ], x\right ) \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 {x^{2} \lambda }{\left (x -1\right )^{2}}+\frac {2 x \lambda }{\left (x -1\right )^{2}}-\frac {\lambda }{\left (x -1\right )^{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*}

Summary of solutions found

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

2.2.17.2 ✓ Maple. Time used: 0.002 (sec). Leaf size: 1087

ode:=a*x^2*(x-1)^2*(diff(y(x),x)+lambda*y(x)^2)+b*x^2+c*x+s = 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) = -(b*x^2+c*x+s)* 
lambda/a/x^2/(x^2-2*x+1)*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 
            <- heuristic approach successful 
         <- hypergeometric successful 
      <- special function solution successful 
   <- Riccati to 2nd Order successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & a \,x^{2} \left (x -1\right )^{2} \left (\frac {d}{d x}y \left (x \right )+\lambda y \left (x \right )^{2}\right )+b \,x^{2}+c x +s =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=-\frac {y \left (x \right )^{2} a \lambda \,x^{4}-2 y \left (x \right )^{2} a \lambda \,x^{3}+a \,x^{2} \lambda y \left (x \right )^{2}+b \,x^{2}+c x +s}{a \,x^{2} \left (x -1\right )^{2}} \end {array} \]

2.2.17.3 ✗ Mathematica

ode=a*x^2*(x-1)^2*(D[y[x],x]+\[Lambda]*y[x]^2)+b*x^2+c*x+s==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Timed out

2.2.17.4 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
lambda_ = symbols("lambda_") 
s = symbols("s") 
y = Function("y") 
ode = Eq(a*x**2*(x - 1)**2*(lambda_*y(x)**2 + Derivative(y(x), x)) + b*x**2 + c*x + s,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out