2.22.9 Problem 11

Internal problem ID [13504]
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 : 11
Date solved : Friday, December 19, 2025 at 05:10:20 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }-y&=-\frac {2 x}{9}+6 A^{2} \left (1+\frac {2 A}{\sqrt {x}}\right ) \\ \end{align*}

Unknown ode type.

2.22.9.1 ✓ Maple. Time used: 0.005 (sec). Leaf size: 354

ode:=y(x)*diff(y(x),x)-y(x) = -2/9*x+6*A^2*(1+2*A/x^(1/2)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-108 A^{3}-54 A^{2} \sqrt {x}+2 x^{{3}/{2}}}{3 \,{\mathrm e}^{\operatorname {RootOf}\left (18 A^{2} {\mathrm e}^{\textit {\_Z}} \ln \left (\frac {\left (3 A -\sqrt {x}\right ) \left (6 A -\sqrt {x}\right ) \left (36 A^{2}-x \right )}{\left (9 A^{2}-x \right ) \left (6 A +\sqrt {x}\right ) \left (3 A +\sqrt {x}\right ) \left ({\mathrm e}^{\textit {\_Z}}+9\right )^{2}}\right )+36 A^{2} {\mathrm e}^{\textit {\_Z}} \ln \left (2\right )+108 A^{2} c_1 \,{\mathrm e}^{\textit {\_Z}}+36 A^{2} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} +3 A \sqrt {x}\, {\mathrm e}^{\textit {\_Z}} \ln \left (\frac {\left (3 A -\sqrt {x}\right ) \left (6 A -\sqrt {x}\right ) \left (36 A^{2}-x \right )}{\left (9 A^{2}-x \right ) \left (6 A +\sqrt {x}\right ) \left (3 A +\sqrt {x}\right ) \left ({\mathrm e}^{\textit {\_Z}}+9\right )^{2}}\right )+6 A \sqrt {x}\, {\mathrm e}^{\textit {\_Z}} \ln \left (2\right )+18 A \sqrt {x}\, c_1 \,{\mathrm e}^{\textit {\_Z}}+6 A \sqrt {x}\, {\mathrm e}^{\textit {\_Z}} \textit {\_Z} +108 A^{2} {\mathrm e}^{\textit {\_Z}}-18 A \sqrt {x}\, {\mathrm e}^{\textit {\_Z}}-x \,{\mathrm e}^{\textit {\_Z}} \ln \left (\frac {\left (3 A -\sqrt {x}\right ) \left (6 A -\sqrt {x}\right ) \left (36 A^{2}-x \right )}{\left (9 A^{2}-x \right ) \left (6 A +\sqrt {x}\right ) \left (3 A +\sqrt {x}\right ) \left ({\mathrm e}^{\textit {\_Z}}+9\right )^{2}}\right )-2 x \,{\mathrm e}^{\textit {\_Z}} \ln \left (2\right )-6 c_1 x \,{\mathrm e}^{\textit {\_Z}}-2 \,{\mathrm e}^{\textit {\_Z}} \textit {\_Z} x +324 A^{2}+54 A \sqrt {x}-18 x \right )} A +9 A +3 \sqrt {x}} \]

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

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=-\frac {2 x}{9}+6 A^{2} \left (1+\frac {2 A}{\sqrt {x}}\right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )-\frac {2 x}{9}+6 A^{2} \left (1+\frac {2 A}{\sqrt {x}}\right )}{y \left (x \right )} \end {array} \]

2.22.9.2 ✓ Mathematica. Time used: 5.749 (sec). Leaf size: 488

ode=y[x]*D[y[x],x]-y[x]==-2/9*x+6*A^2*(1+2*A*x^(-1/2)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {2^{2/3} \left (\frac {-\frac {6 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2}{y(x)}-9 \sqrt {x}}{\sqrt [3]{A^3}}+54\right ) \left (\frac {6 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2+9 \sqrt {x} y(x)}{\sqrt [3]{A^3} y(x)}+27\right ) \left (-\frac {\left (3 \left (3 \sqrt [3]{A^3}+\sqrt {x}\right ) y(x)+2 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2\right ) \log \left (\frac {1}{27} 2^{2/3} \left (\frac {-\frac {6 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2}{y(x)}-9 \sqrt {x}}{\sqrt [3]{A^3}}+54\right )\right )}{9 \sqrt [3]{A^3} y(x)}+\left (\frac {2 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2+3 \sqrt {x} y(x)}{9 \sqrt [3]{A^3} y(x)}+1\right ) \log \left (\frac {1}{27} 2^{2/3} \left (\frac {6 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2+9 \sqrt {x} y(x)}{\sqrt [3]{A^3} y(x)}+27\right )\right )-3\right )}{6561 \left (\frac {\left (2 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2+3 \sqrt {x} y(x)\right )^3}{729 A^3 y(x)^3}+\frac {-\frac {6 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2}{y(x)}-9 \sqrt {x}}{9 \sqrt [3]{A^3}}-2\right )}=\frac {2^{2/3} \left (A^3\right )^{2/3} \left (2 \text {arctanh}\left (\frac {1}{3}-\frac {2 \sqrt {x}}{9 A}\right )+\frac {9 A}{3 A+\sqrt {x}}\right )}{9 A^2}+c_1,y(x)\right ] \]

2.22.9.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
y = Function("y") 
ode = Eq(-6*A**2*(2*A/sqrt(x) + 1) + 2*x/9 + y(x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out