2.22.22 Problem 26

Internal problem ID [13517]
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 : 26
Date solved : Friday, December 19, 2025 at 05:29:49 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }-y&=-\frac {2 x}{9}+\frac {A}{\sqrt {x}} \\ \end{align*}

Unknown ode type.

2.22.22.1 ✓ Maple. Time used: 0.006 (sec). Leaf size: 134

ode:=y(x)*diff(y(x),x)-y(x) = -2/9*x+A/x^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {2 \,3^{{5}/{6}} 2^{{1}/{3}} \left (-2 x^{{3}/{2}}+9 A \right )}{\sqrt {x}\, \left (\left (27 \tan \left (\operatorname {RootOf}\left (18 \,3^{{5}/{6}} 2^{{1}/{3}} \int \frac {\left (\frac {A}{x^{{3}/{2}}}\right )^{{2}/{3}} \sqrt {x}}{-2 x^{{3}/{2}}+9 A}d x +\sqrt {3}\, \ln \left (-8 \sin \left (\textit {\_Z} \right ) \sqrt {3}\, \cos \left (\textit {\_Z} \right )^{3}-8 \cos \left (\textit {\_Z} \right )^{4}-4 \sin \left (\textit {\_Z} \right ) \cos \left (\textit {\_Z} \right ) \sqrt {3}+16 \cos \left (\textit {\_Z} \right )^{2}+1\right )-12 \sqrt {3}\, c_1 -12 \textit {\_Z} \right )\right )-9 \sqrt {3}\right ) \left (\frac {A}{x^{{3}/{2}}}\right )^{{1}/{3}}-6 \,2^{{1}/{3}} 3^{{5}/{6}}\right )} \]

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

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

2.22.22.2 ✓ Mathematica. Time used: 0.467 (sec). Leaf size: 282

ode=y[x]*D[y[x],x]-y[x]==-2/9*x+A*x^(-1/2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\log \left (9 A^{2/3}+3 \sqrt [3]{6} \sqrt [3]{A} \sqrt {x}+6^{2/3} x\right )+2 \sqrt {3} \arctan \left (\frac {-\frac {6 \sqrt [3]{6} \left (9 A-2 x^{3/2}+3 \sqrt {x} y(x)\right )}{\sqrt [3]{A} y(x)}-27}{27 \sqrt {3}}\right )+2 \sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{6} \sqrt {x}}{\sqrt [3]{A}}+3}{3 \sqrt {3}}\right )+2 \log \left (\frac {1}{27} \left (27-\frac {3 \sqrt [3]{6} \left (9 A-2 x^{3/2}+3 \sqrt {x} y(x)\right )}{\sqrt [3]{A} y(x)}\right )\right )=\log \left (\frac {1}{81} \left (\frac {6^{2/3} \left (9 A-2 x^{3/2}+3 \sqrt {x} y(x)\right )^2}{A^{2/3} y(x)^2}+\frac {9 \sqrt [3]{6} \left (9 A-2 x^{3/2}+3 \sqrt {x} y(x)\right )}{\sqrt [3]{A} y(x)}+81\right )\right )+2 \log \left (3 \sqrt [3]{A}-\sqrt [3]{6} \sqrt {x}\right )+6 c_1,y(x)\right ] \]

2.22.22.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
y = Function("y") 
ode = Eq(-A/sqrt(x) + 2*x/9 + 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