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2.25.21 Problem 38

2.25.21.1 ✗Maple
2.25.21.2 ✗Mathematica
2.25.21.3 ✗Sympy

Internal problem ID [13636]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.4-2.
Problem number : 38
Date solved : Friday, December 19, 2025 at 09:42:01 AM
CAS classification : [[_Abel, `2nd type`, `class A`]]

\begin{align*} y y^{\prime }&=-n y^{2}+a \left (1+2 n \right ) {\mathrm e}^{x} y+b y-a^{2} n \,{\mathrm e}^{2 x}-{\mathrm e}^{x} a b +c \\ \end{align*}
Unknown ode type.
2.25.21.1 ✗ Maple
ode:=y(x)*diff(y(x),x) = -n*y(x)^2+a*(1+2*n)*exp(x)*y(x)+b*y(x)-a^2*n*exp(2*x)-a*b*exp(x)+c; 
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), 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)*exp(x)*a*(2*n+1)/(2*a*exp 
(x)*n+a*exp(x)+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)+(2*y(x)*exp(2*x)*a^2*n+y(x)* 
exp(x)*a*b+2*n)/(a^2*n*exp(2*x)+a*b*exp(x)-c), y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Computing symmetries using: way = HINT 
   -> Calling odsolve with the ODE, diff(y(x),x) = -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)*b/x/(-2*n*x+b-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)+(2*y(x)*exp(2*x)*a^2*n+y(x)* 
exp(x)*a*b+2*n*K[1])/(a^2*n*exp(2*x)+a*b*exp(x)-c), 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)+(2*n+1)*(y(x)*exp(x)*a-n*K[1]) 
/(2*a*exp(x)*n+a*exp(x)+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)+(2*y(x)*exp(2*x)*a^2*n+y(x)* 
exp(x)*a*b-2*b*n*K[1])/(a^2*n*exp(2*x)+a*b*exp(x)-c), 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) = 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) = -y(x)*exp(x)*a*(2*n+1)/(2*a* 
exp(x)*n+a*exp(x)+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)+(-2*K[1]*x+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)*(n*x^2+c)/x/(-n*x^2+b*x+c 
), 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)+(2*K[1]*x^2*n-K[1]*b*x+K[1]*x^ 
2+y(x)*b)/x/(-2*n*x+b-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)/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)+(4*y(x)*b*n-n*K[1]*x+y(x)*b)/( 
-4*n^2*x+4*b*n-4*n*x+b-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)+(-4*y(x)*b*n^2*x^2+2*K[1]*n^2* 
x^3-2*y(x)*b*n*x^2+n*K[1]*x^3+y(x)*b^3-4*y(x)*b*c*n-b^2*x*K[1]-2*c*n*x*K[1]-2*y 
(x)*b*c-c*x*K[1])/b/x/(4*n^2*x^2-6*b*n*x+2*n*x^2+b^2-3*b*x-4*c*n-2*c), 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)+(-4*y(x)*n^3*x^2-4*y(x)*n^2*x^ 
2+5*y(x)*b^2*n-4*y(x)*c*n^2-y(x)*n*x^2-2*b*n*K[1]*x+2*y(x)*b^2-4*y(x)*c*n-y(x)* 
c)/x/(4*n^3*x^2-12*b*n^2*x+4*n^2*x^2+5*b^2*n-12*b*n*x-4*c*n^2+n*x^2+2*b^2-3*b*x 
-4*c*n-c), 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

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )=-13636 y \left (x \right )^{2}+27273 a \,{\mathrm e}^{x} y \left (x \right )+b y \left (x \right )-13636 a^{2} {\mathrm e}^{2 x}-a b \,{\mathrm e}^{x}+c \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {-13636 y \left (x \right )^{2}+27273 a \,{\mathrm e}^{x} y \left (x \right )+b y \left (x \right )-13636 a^{2} {\mathrm e}^{2 x}-a b \,{\mathrm e}^{x}+c}{y \left (x \right )} \end {array} \]
2.25.21.2 ✗ Mathematica
ode=y[x]*D[y[x],x]==-n*y[x]^2+a*(2*n+1)*Exp[x]*y[x]+b*y[x]-a^2*n*Exp[2*x]-a*b*Exp[x]+c; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]

Not solved

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

NotImplementedError : The given ODE -(-a**2*n*exp(2*x) - a*b*exp(x) - c + (2*a*n*exp(x) + a*exp(x) +
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0
 

classify_ode(ode,func=y(x)) 
 
('factorable', '1st_power_series', 'lie_group')

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