55.23.5 problem 5

Internal problem ID [13622]
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 1.3.2.
Problem number : 5
Date solved : Sunday, October 12, 2025 at 04:13:50 AM
CAS classification : [[_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }&=\frac {3 y}{\sqrt {a \,x^{{3}/{2}}+8 x}}+1 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 293

ode:=y(x)*diff(y(x),x) = 3/(a*x^(3/2)+8*x)^(1/2)*y(x)+1; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {{\left (-\frac {a \sqrt {x}\, \left (-2 a \,x^{{3}/{2}}+\sqrt {x}\, a y^{2}-8 \sqrt {x \left (8+\sqrt {x}\, a \right )}\, y-16 x \right )}{\left (\sqrt {x}\, a y-4 \sqrt {x \left (8+\sqrt {x}\, a \right )}\right )^{2}}\right )}^{{1}/{4}} \sqrt {2 \sqrt {x}\, a +16}\, a \sqrt {x}\, y+4 \left (\sqrt {x}\, a y-4 \sqrt {x \left (8+\sqrt {x}\, a \right )}\right ) \left (\int _{}^{-\frac {\sqrt {2 \sqrt {x}\, a +16}\, \sqrt {x \left (8+\sqrt {x}\, a \right )}}{\sqrt {x}\, a y-4 \sqrt {x \left (8+\sqrt {x}\, a \right )}}}\frac {\left (\textit {\_a}^{2}-1\right )^{{1}/{4}}}{\sqrt {\textit {\_a}}}d \textit {\_a} +\frac {c_1}{4}\right ) \sqrt {-\frac {\sqrt {2 \sqrt {x}\, a +16}\, \sqrt {x \left (8+\sqrt {x}\, a \right )}}{\sqrt {x}\, a y-4 \sqrt {x \left (8+\sqrt {x}\, a \right )}}}}{\sqrt {-\frac {\sqrt {2 \sqrt {x}\, a +16}\, \sqrt {x \left (8+\sqrt {x}\, a \right )}}{\sqrt {x}\, a y-4 \sqrt {x \left (8+\sqrt {x}\, a \right )}}}\, \left (\sqrt {x}\, a y-4 \sqrt {x \left (8+\sqrt {x}\, a \right )}\right )} = 0 \]

✗ Mathematica

ode=y[x]*D[y[x],x]==3*(a*x^(3/2)+8*x)^(-1/2)*y[x]+1; 
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(y(x)*Derivative(y(x), x) - 1 - 3*y(x)/sqrt(a*x**(3/2) + 8*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - 1/y(x) - 3/sqrt(a*x**(3/2) + 8*x) cannot be solved by the factorable group method