2.22.58 Problem 76

Internal problem ID [13553]
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 1.3.1-2. Solvable equations and their solutions
Problem number : 76
Date solved : Friday, December 19, 2025 at 06:50:14 AM
CAS classification : [[_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }-y&=a^{2} f^{\prime }\left (x \right ) f^{\prime \prime }\left (x \right )-\frac {\left (f \left (x \right )+b \right )^{2} f^{\prime \prime }\left (x \right )}{{f^{\prime }\left (x \right )}^{3}} \\ \end{align*}

Unknown ode type.

2.22.58.1 ✗ Maple

ode:=y(x)*diff(y(x),x)-y(x) = a^2*diff(f(x),x)*diff(diff(f(x),x),x)-(f(x)+b)^2/diff(f(x),x)^3*diff(diff(f(x),x),x); 
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)*(a^2*diff(f(x),x)^5*diff( 
diff(diff(f(x),x),x),x)+a^2*diff(diff(f(x),x),x)^2*diff(f(x),x)^4-diff(diff( 
diff(f(x),x),x),x)*diff(f(x),x)*f(x)^2-2*diff(diff(diff(f(x),x),x),x)*diff(f(x) 
,x)*f(x)*b-diff(diff(diff(f(x),x),x),x)*diff(f(x),x)*b^2-2*diff(diff(f(x),x),x) 
*diff(f(x),x)^2*f(x)-2*diff(diff(f(x),x),x)*diff(f(x),x)^2*b+3*diff(diff(f(x),x 
),x)^2*f(x)^2+6*diff(diff(f(x),x),x)^2*f(x)*b+3*diff(diff(f(x),x),x)^2*b^2)/ 
diff(f(x),x)/diff(diff(f(x),x),x)/(a*diff(f(x),x)^2-f(x)-b)/(a*diff(f(x),x)^2+f 
(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 
   -> 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)*a^2+2*b*x)/a^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 
-> 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
 

2.22.58.2 ✗ Mathematica

ode=y[x]*D[y[x],x]-y[x]==a^2*D[ f[x],x]*D[ f[x],{x,2}]-(f[x]+b)^2/( (D[ f[x],x])^3)*D[ f[x],{x,2}]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.22.58.3 ✗ Sympy

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

Timed Out