55.22.34 problem 34

Internal problem ID [13575]
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 : 34
Date solved : Sunday, October 12, 2025 at 03:48:56 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }-y&=A \left (n +2\right ) \left (\sqrt {x}+2 \left (n +2\right ) A +\frac {\left (n +1\right ) \left (n +3\right ) A^{2}}{\sqrt {x}}\right ) \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 410

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

\[ \frac {\left (n +2\right ) \left (\operatorname {BesselI}\left (\frac {n +3}{n +2}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (n^{2}+4 n +3\right ) A^{2}+x -y}{\left (n +2\right )^{2} A^{2}}}\right ) c_1 +\operatorname {BesselK}\left (\frac {n +3}{n +2}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (n^{2}+4 n +3\right ) A^{2}+x -y}{\left (n +2\right )^{2} A^{2}}}\right )\right ) A \sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (n^{2}+4 n +3\right ) A^{2}+x -y}{\left (n +2\right )^{2} A^{2}}}+\left (c_1 \operatorname {BesselI}\left (\frac {1}{n +2}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (n^{2}+4 n +3\right ) A^{2}+x -y}{\left (n +2\right )^{2} A^{2}}}\right )-\operatorname {BesselK}\left (\frac {1}{n +2}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (n^{2}+4 n +3\right ) A^{2}+x -y}{\left (n +2\right )^{2} A^{2}}}\right )\right ) \left (\sqrt {x}+\left (n +1\right ) A \right )}{A \sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (n^{2}+4 n +3\right ) A^{2}+x -y}{\left (n +2\right )^{2} A^{2}}}\, \left (n +2\right ) \operatorname {BesselI}\left (\frac {n +3}{n +2}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (n^{2}+4 n +3\right ) A^{2}+x -y}{\left (n +2\right )^{2} A^{2}}}\right )+\left (\sqrt {x}+\left (n +1\right ) A \right ) \operatorname {BesselI}\left (\frac {1}{n +2}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+\left (n^{2}+4 n +3\right ) A^{2}+x -y}{\left (n +2\right )^{2} A^{2}}}\right )} = 0 \]

✗ Mathematica

ode=y[x]*D[y[x],x]-y[x]==A*(n+2)*(x^(1/2)+2*(n+2)*A+(n+1)*(n+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") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-A*(n + 2)*(A**2*(n + 1)*(n + 3)/sqrt(x) + A*(2*n + 4) + sqrt(x)) + y(x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out