2.22.36 Problem 42

Internal problem ID [13531]
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 : 42
Date solved : Friday, December 19, 2025 at 06:00:03 AM
CAS classification : [[_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }-y&=\frac {9 x}{32}+\frac {15 \sqrt {b^{2}+x^{2}}}{32}+\frac {3 b^{2}}{64 \sqrt {b^{2}+x^{2}}} \\ \end{align*}

Unknown ode type.

2.22.36.1 ✗ Maple

ode:=y(x)*diff(y(x),x)-y(x) = 9/32*x+15/32*(b^2+x^2)^(1/2)+3/64*b^2/(b^2+x^2)^(1/2); 
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), 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)*(6*(b^2+x^2)^(1/2)*b^2+6* 
(b^2+x^2)^(1/2)*x^2+9*b^2*x+10*x^3)/(b^2+x^2)/(11*b^2+10*x^2+6*x*(b^2+x^2)^(1/2 
)), 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)+1/5*(5*y(x)-6*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)+1/3*(3*y(x)-10*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)+1/11*(11*y(x)-6*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 
-> 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

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=\frac {9 x}{32}+\frac {15 \sqrt {b^{2}+x^{2}}}{32}+\frac {3 b^{2}}{64 \sqrt {b^{2}+x^{2}}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )+\frac {9 x}{32}+\frac {15 \sqrt {b^{2}+x^{2}}}{32}+\frac {3 b^{2}}{64 \sqrt {b^{2}+x^{2}}}}{y \left (x \right )} \end {array} \]

2.22.36.2 ✗ Mathematica

ode=y[x]*D[y[x],x]-y[x]==9/32*x+15/32*Sqrt[x^2+b^2]+3*b^2/(64*Sqrt[x^2+b^2]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.22.36.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
b = symbols("b") 
y = Function("y") 
ode = Eq(-3*b**2/(64*sqrt(b**2 + x**2)) - 9*x/32 - 15*sqrt(b**2 + x**2)/32 + y(x)*Derivative(y(x), x) - y(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