61.22.37 problem 37

Internal problem ID [12283]
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 : 37
Date solved : Monday, March 31, 2025 at 05:02:29 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 228

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

\[ \frac {\left (-2 A +\sqrt {x}\right ) \operatorname {BesselK}\left (1, -\sqrt {-\frac {2 A \sqrt {x}-x +y}{A^{2}}}\right )+\operatorname {BesselK}\left (0, -\sqrt {-\frac {2 A \sqrt {x}-x +y}{A^{2}}}\right ) \sqrt {-\frac {2 A \sqrt {x}-x +y}{A^{2}}}\, A +c_1 \left (A \operatorname {BesselI}\left (0, \sqrt {-\frac {2 A \sqrt {x}-x +y}{A^{2}}}\right ) \sqrt {-\frac {2 A \sqrt {x}-x +y}{A^{2}}}+\left (-2 A +\sqrt {x}\right ) \operatorname {BesselI}\left (1, \sqrt {-\frac {2 A \sqrt {x}-x +y}{A^{2}}}\right )\right )}{A \operatorname {BesselI}\left (0, \sqrt {-\frac {2 A \sqrt {x}-x +y}{A^{2}}}\right ) \sqrt {-\frac {2 A \sqrt {x}-x +y}{A^{2}}}+\left (-2 A +\sqrt {x}\right ) \operatorname {BesselI}\left (1, \sqrt {-\frac {2 A \sqrt {x}-x +y}{A^{2}}}\right )} = 0 \]

✗ Mathematica

ode=y[x]*D[y[x],x]-y[x]==2*A^2-A*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(-2*A**2 + A*sqrt(x) + y(x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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