55.22.17 problem 17

Internal problem ID [13558]
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 : 17
Date solved : Sunday, October 12, 2025 at 03:42:08 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }-y&=-\frac {x}{4}+\frac {A \left (\sqrt {x}+5 A +\frac {3 A^{2}}{\sqrt {x}}\right )}{4} \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 209

ode:=y(x)*diff(y(x),x)-y(x) = -1/4*x+1/4*A*(x^(1/2)+5*A+3*A^2/x^(1/2)); 
dsolve(ode,y(x), singsol=all);
 

\[ c_1 +\frac {\left (-\sqrt {x}+3 A \right ) \sqrt {\frac {3 A^{2}+2 \sqrt {x}\, A -x +2 y}{6 A^{2}-2 \sqrt {x}\, A +y}}\, y \,{\mathrm e}^{\frac {-6 A^{2}+2 \sqrt {x}\, A -y}{3 A^{2}+2 \sqrt {x}\, A -x +2 y}}}{\sqrt {-\frac {\left (-\sqrt {x}+3 A \right )^{2}}{6 A^{2}-2 \sqrt {x}\, A +y}}\, \left (12 A^{3}-4 A^{2} \sqrt {x}+2 y A \right )}-\int _{}^{\frac {6 \sqrt {x}\, A -2 x +3 y}{12 A^{2}-4 \sqrt {x}\, A +2 y}}\frac {{\mathrm e}^{-\frac {2}{2 \textit {\_a} +1}} \sqrt {2 \textit {\_a} +1}}{\sqrt {2 \textit {\_a} -3}}d \textit {\_a} = 0 \]

✗ Mathematica

ode=y[x]*D[y[x],x]-y[x]==-1/4*x+1/4*A*(x^(1/2)+5*A+3*A^2*x^(-1/2)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
y = Function("y") 
ode = Eq(-A*(3*A**2/sqrt(x) + 5*A + sqrt(x))/4 + x/4 + y(x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (3*A**3 + sqrt(x)*(5*A**2 + A*sqrt(x) - x) + 4*sqrt(x)*y(x))/(4*sqrt(x)*y(x)) cannot be solved by the factorable group method