61.22.36 problem 36

Internal problem ID [12282]
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 : 36
Date solved : Friday, March 14, 2025 at 04:42:06 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 407

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

\[ \frac {-c_{1} \left (\sqrt {\frac {A^{3}-B}{A^{3}}}\, A -A -\sqrt {x}\right ) \operatorname {BesselI}\left (\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}-y A +x A +B}{A^{3}}}\right )+A \operatorname {BesselI}\left (1+\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}-y A +x A +B}{A^{3}}}\right ) \sqrt {\frac {2 A^{2} \sqrt {x}-y A +x A +B}{A^{3}}}\, c_{1} +\operatorname {BesselK}\left (1+\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}-y A +x A +B}{A^{3}}}\right ) \sqrt {\frac {2 A^{2} \sqrt {x}-y A +x A +B}{A^{3}}}\, A +\left (\sqrt {\frac {A^{3}-B}{A^{3}}}\, A -A -\sqrt {x}\right ) \operatorname {BesselK}\left (\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}-y A +x A +B}{A^{3}}}\right )}{\left (-\sqrt {\frac {A^{3}-B}{A^{3}}}\, A +A +\sqrt {x}\right ) \operatorname {BesselI}\left (\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}-y A +x A +B}{A^{3}}}\right )+\operatorname {BesselI}\left (1+\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}-y A +x A +B}{A^{3}}}\right ) \sqrt {\frac {2 A^{2} \sqrt {x}-y A +x A +B}{A^{3}}}\, A} = 0 \]

✗ Mathematica

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

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
B = symbols("B") 
y = Function("y") 
ode = Eq(-2*A**2 - A*sqrt(x) - B/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/y(x) - A*sqrt(x)/y(x) - B/(sqrt(x)*y(x)) + Derivative(y(x), x) - 1 cannot be solved by the factorable group method