55.22.3 problem 3

Internal problem ID [13544]
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 : 3
Date solved : Sunday, October 12, 2025 at 03:36:09 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 90

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

\[ y = -\frac {9 A \left (A \sqrt {x}+B -\frac {2 x^{{3}/{2}}}{9}\right )}{3 A \sqrt {x}+3 \operatorname {RootOf}\left (18 A^{3} \int _{}^{\textit {\_Z}}\frac {1}{-2 \textit {\_a}^{3} B^{2}+9 \textit {\_a} \,A^{3}-9 A^{3}}d \textit {\_a} -9 A \int \frac {1}{9 A x -2 x^{2}+9 B \sqrt {x}}d x +2 c_1 \right ) B} \]

✓ Mathematica. Time used: 4.759 (sec). Leaf size: 415

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

\[ \text {Solve}\left [6 \text {RootSum}\left [8 \text {$\#$1}^6-72 \text {$\#$1}^4 A-36 \text {$\#$1}^4 y(x)-72 \text {$\#$1}^3 B+162 \text {$\#$1}^2 A^2+162 \text {$\#$1}^2 A y(x)+54 \text {$\#$1}^2 y(x)^2+324 \text {$\#$1} A B+162 \text {$\#$1} B y(x)-81 A y(x)^2+162 B^2-27 y(x)^3\&,\frac {-2 \text {$\#$1}^3 \log \left (\sqrt {x}-\text {$\#$1}\right )+9 \text {$\#$1} A \log \left (\sqrt {x}-\text {$\#$1}\right )+9 B \log \left (\sqrt {x}-\text {$\#$1}\right )+9 \text {$\#$1} y(x) \log \left (\sqrt {x}-\text {$\#$1}\right )}{8 \text {$\#$1}^5-48 \text {$\#$1}^3 A-24 \text {$\#$1}^3 y(x)-36 \text {$\#$1}^2 B+54 \text {$\#$1} A^2+54 \text {$\#$1} A y(x)+18 \text {$\#$1} y(x)^2+54 A B+27 B y(x)}\&\right ]+\int _1^{y(x)}\left (\frac {162 K[1]}{8 x^3-72 A x^2-36 K[1] x^2-72 B x^{3/2}+162 A^2 x+54 K[1]^2 x+162 A K[1] x+324 A B \sqrt {x}+162 B K[1] \sqrt {x}-27 K[1]^3+162 B^2-81 A K[1]^2}+\frac {162 K[1]}{-8 x^3+72 A x^2+36 K[1] x^2+72 B x^{3/2}-162 A^2 x-54 K[1]^2 x-162 A K[1] x-324 A B \sqrt {x}-162 B K[1] \sqrt {x}+27 K[1]^3-162 B^2+81 A K[1]^2}\right )dK[1]=c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out