2.26.12 Problem 12

Internal problem ID [13648]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.4. Equations Containing Polynomial Functions of y. subsection 1.4.1-2 Abel equations of the first kind.
Problem number : 12
Date solved : Friday, December 19, 2025 at 09:57:41 AM
CAS classification : [_Abel]

\begin{align*} 9 y^{\prime }&=-x^{m} \left (a \,x^{-m +1}+b \right )^{2 \lambda +1} y^{3}-x^{-2 m} \left (9 a +2+9 b m \,x^{m -1}\right ) \left (a \,x^{-m +1}+b \right )^{-\lambda -2} \\ \end{align*}

Unknown ode type.

2.26.12.1 ✗ Maple

ode:=9*diff(y(x),x) = -x^m*(a*x^(-m+1)+b)^(2*lambda+1)*y(x)^3-x^(-2*m)*(9*a+2+9*b*m*x^(m-1))*(a*x^(-m+1)+b)^(-lambda-2); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

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 
trying Abel 
Looking for potential symmetries 
Looking for potential symmetries 
Looking for potential symmetries 
trying inverse_Riccati 
trying an equivalence to an Abel ODE 
differential order: 1; trying a linearization to 2nd order 
--- trying a change of variables {x -> y(x), y(x) -> x} 
differential order: 1; trying a linearization to 2nd order 
trying 1st order ODE linearizable_by_differentiation 
--- Trying Lie symmetry methods, 1st order --- 
   -> Computing symmetries using: way = 3 
   -> Computing symmetries using: way = 4 
   -> Computing symmetries using: way = 2 
trying symmetry patterns for 1st order ODEs 
-> trying a symmetry pattern of the form [F(x)*G(y), 0] 
-> trying a symmetry pattern of the form [0, F(x)*G(y)] 
-> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] 
   -> Computing symmetries using: way = HINT 
   -> Calling odsolve with the ODE, diff(y(x),x) = 0, y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x) = 3*y(x)/x, y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x)-y(x)*(2*x^(1-m)*a*lambda*m-2*x 
^(1-m)*a*lambda-a*x^(1-m)-b*m)/x/(a*x^(1-m)+b), y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x)+y(x)*(9*x^(m-1)*x^(1-m)*a*b* 
lambda*m^2-9*x^(m-1)*x^(1-m)*a*b*lambda*m+9*x^(m-1)*x^(1-m)*a*b*m^2-27*x^(m-1)* 
x^(1-m)*a*b*m-9*x^(m-1)*b^2*m^2+9*x^(1-m)*a^2*lambda*m-9*x^(m-1)*b^2*m-9*x^(1-m 
)*a^2*lambda+2*x^(1-m)*a*lambda*m-18*x^(1-m)*a^2-2*x^(1-m)*a*lambda-18*a*b*m-4* 
a*x^(1-m)-4*b*m)/x/(9*a+2+9*b*m*x^(m-1))/(a*x^(1-m)+b), y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x) = y(x)*(2*x^(1-m)*a*lambda*m-2 
*x^(1-m)*a*lambda-a*x^(1-m)-b*m)/x/(a*x^(1-m)+b), y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
-> trying a symmetry pattern of the form [F(x),G(x)] 
-> trying a symmetry pattern of the form [F(y),G(y)] 
-> trying a symmetry pattern of the form [F(x)+G(y), 0] 
-> trying a symmetry pattern of the form [0, F(x)+G(y)] 
-> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] 
-> trying a symmetry pattern of conformal type
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 9 \frac {d}{d x}y \left (x \right )=-x^{m} \left (a \,x^{1-m}+b \right )^{2 \lambda +1} y \left (x \right )^{3}-x^{-2 m} \left (9 a +2+9 b m \,x^{m -1}\right ) \left (a \,x^{1-m}+b \right )^{-\lambda -2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=-\frac {x^{m} \left (a \,x^{1-m}+b \right )^{2 \lambda +1} y \left (x \right )^{3}}{9}-\frac {x^{-2 m} \left (9 a +2+9 b m \,x^{m -1}\right ) \left (a \,x^{1-m}+b \right )^{-\lambda -2}}{9} \end {array} \]

2.26.12.2 ✗ Mathematica

ode=9*D[y[x],x]==-x^m*(a*x^(1-m)+b)^(2*\[Lambda]+1)*y[x]^3-x^(-2*m)*(9*a+2+9*b*m*x^(m-1))*(a*x^(1-m)+b)^(-\[Lambda]-2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.26.12.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
lambda_ = symbols("lambda_") 
m = symbols("m") 
y = Function("y") 
ode = Eq(x**m*(a*x**(1 - m) + b)**(2*lambda_ + 1)*y(x)**3 + 9*Derivative(y(x), x) + (a*x**(1 - m) + b)**(-lambda_ - 2)*(9*a + 9*b*m*x**(m - 1) + 2)/x**(2*m),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out