2.22.4 Problem 4

Internal problem ID [13499]
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 : 4
Date solved : Friday, December 19, 2025 at 05:06:26 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }-y&=2 A \left (\sqrt {x}+4 A +\frac {3 A^{2}}{\sqrt {x}}\right ) \\ \end{align*}

Unknown ode type.

2.22.4.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 120

ode:=y(x)*diff(y(x),x)-y(x) = 2*A*(x^(1/2)+4*A+3*A^2/x^(1/2)); 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {4 \,\operatorname {arctanh}\left (\frac {\sqrt {-\frac {A^{2}}{y}}\, \left (3 A +\sqrt {x}\right )}{\sqrt {\frac {-3 A^{2}-4 A \sqrt {x}-x +y}{y}}\, A}\right ) \sqrt {-\frac {A^{2}}{y}}+c_1 \sqrt {-\frac {A^{2}}{y}}-\sqrt {\frac {-6 A^{2}-8 A \sqrt {x}-2 x +2 y}{y}}\, \sqrt {2}}{\sqrt {-\frac {A^{2}}{y}}} = 0 \]

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 
<- Abel successful
 

Maple step by step

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

2.22.4.2 ✗ Mathematica

ode=y[x]*D[y[x],x]-y[x]==2*A*(x^(1/2)+4*A+3*A^2*x^(-1/2)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.22.4.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
y = Function("y") 
ode = Eq(-2*A*(3*A**2/sqrt(x) + 4*A + sqrt(x)) + y(x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out