61.22.17 problem 17

Internal problem ID [12263]
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 : Friday, March 14, 2025 at 04:39:52 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: 0.003 (sec). Leaf size: 257

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);
 

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

✗ 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