61.22.19 problem 19

Internal problem ID [12265]
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 : 19
Date solved : Friday, March 14, 2025 at 04:40:08 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 170

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

\[ \frac {6 \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {x \left (A^{2}\right )^{{1}/{3}}}{A}}\, \left (y-2 x \right )}{\sqrt {\frac {\left (4 x^{3}-4 x^{2} y+x y^{2}+2 A \right ) \left (A^{2}\right )^{{1}/{3}}}{y^{2} A}}\, y}\right ) A +\frac {c_{1}}{6}\right ) x \sqrt {\frac {x \left (A^{2}\right )^{{1}/{3}}}{A}}+2 y \sqrt {3}\, \left (-x^{3}-\frac {x^{2} y}{2}+\frac {x y^{2}}{2}+A \right ) \sqrt {\frac {\left (4 x^{3}-4 x^{2} y+x y^{2}+2 A \right ) \left (A^{2}\right )^{{1}/{3}}}{y^{2} A}}}{\sqrt {\frac {x \left (A^{2}\right )^{{1}/{3}}}{A}}\, x} = 0 \]

✓ Mathematica. Time used: 1.451 (sec). Leaf size: 233

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

\[ \text {Solve}\left [c_1=-\frac {i \sqrt {-\frac {2 A+4 x^3-4 x^2 y(x)+x y(x)^2}{A}} \left (-6 \sqrt {A} x^{3/2} \text {arcsinh}\left (\frac {\sqrt {x} (2 x-y(x))}{\sqrt {2} \sqrt {A}}\right )+x^2 (-y(x)) \sqrt {\frac {2 A+4 x^3-4 x^2 y(x)+x y(x)^2}{A}}+x y(x)^2 \sqrt {\frac {2 A+4 x^3-4 x^2 y(x)+x y(x)^2}{A}}+2 \left (A-x^3\right ) \sqrt {\frac {2 A+4 x^3-4 x^2 y(x)+x y(x)^2}{A}}\right )}{4 \sqrt {A} x^{3/2} \sqrt {\frac {2 A+4 x^3-4 x^2 y(x)+x y(x)^2}{A}}},y(x)\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE -A/(x**2*y(x)) - 2*x/y(x) + Derivative(y(x), x) - 1 cannot be solved by the factorable group method