2.24.45 Problem 71

Internal problem ID [13609]
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 1.3.3-2.
Problem number : 71
Date solved : Friday, December 19, 2025 at 08:34:01 AM
CAS classification : [[_Abel, `2nd type`, `class A`]]

\begin{align*} y y^{\prime }&={\mathrm e}^{x a} \left (2 a \,x^{2}+b +2 x \right ) y+{\mathrm e}^{2 x a} \left (-a \,x^{4}-b \,x^{2}+c \right ) \\ \end{align*}

Unknown ode type.

2.24.45.1 ✗ Maple

ode:=y(x)*diff(y(x),x) = exp(a*x)*(2*a*x^2+b+2*x)*y(x)+exp(2*a*x)*(-a*x^4-b*x^2+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)*(2*a^2*x^2+a*b+6*a*x+2)/( 
2*a*x^2+b+2*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)*(a^2*x^4+a*b*x^2+2*a*x^ 
3-a*c+b*x)/(a*x^4+b*x^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 
   -> 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) = -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*a^2*x^2+a*b+6*a*x+2 
)/(2*a*x^2+b+2*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)+1/4*K[1]/x, y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      <- quadrature successful 
   -> Calling odsolve with the ODE, diff(y(x),x)-c/x*K[1]*(a*b-2)/b^3, y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      <- quadrature successful 
   -> Calling odsolve with the ODE, diff(y(x),x)+3/4*a*b/x*K[1]/(3*a*b+1), y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      <- quadrature successful 
   -> Calling odsolve with the ODE, diff(y(x),x)+1/3/x*K[1]*(a*c+b^2)/b^2, y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      <- quadrature successful 
   -> Calling odsolve with the ODE, diff(y(x),x)-1/6/x*K[1]*(2*a^2*c-a*b^2-2*b) 
/b/(a*b+2), y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      <- quadrature 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

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

2.24.45.2 ✗ Mathematica

ode=y[x]*D[y[x],x]==Exp[a*x]*(2*a*x^2+2*x+b)*y[x]+Exp[2*a*x]*(-a*x^4-b*x^2+c); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.24.45.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq((-2*a*x**2 - b - 2*x)*y(x)*exp(a*x) - (-a*x**4 - b*x**2 + c)*exp(2*a*x) + y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out
 

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