2.23.5 Problem 5
Internal
problem
ID
[13558]
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
:
Friday, December 19, 2025 at 06:54:20 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*}
Unknown ode type.
2.23.5.1 ✓ Maple. Time used: 0.004 (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);
\[
\frac {{\left (-\frac {a \sqrt {x}\, \left (-2 a \,x^{{3}/{2}}+\sqrt {x}\, a y^{2}-8 \sqrt {x \left (a \sqrt {x}+8\right )}\, y-16 x \right )}{\left (\sqrt {x}\, a y-4 \sqrt {x \left (a \sqrt {x}+8\right )}\right )^{2}}\right )}^{{1}/{4}} \sqrt {2 a \sqrt {x}+16}\, a \sqrt {x}\, y+4 \left (\sqrt {x}\, a y-4 \sqrt {x \left (a \sqrt {x}+8\right )}\right ) \sqrt {-\frac {\sqrt {2 a \sqrt {x}+16}\, \sqrt {x \left (a \sqrt {x}+8\right )}}{\sqrt {x}\, a y-4 \sqrt {x \left (a \sqrt {x}+8\right )}}}\, \left (\int _{}^{-\frac {\sqrt {2 a \sqrt {x}+16}\, \sqrt {x \left (a \sqrt {x}+8\right )}}{\sqrt {x}\, a y-4 \sqrt {x \left (a \sqrt {x}+8\right )}}}\frac {\left (\textit {\_a}^{2}-1\right )^{{1}/{4}}}{\sqrt {\textit {\_a}}}d \textit {\_a} +\frac {c_1}{4}\right )}{\sqrt {-\frac {\sqrt {2 a \sqrt {x}+16}\, \sqrt {x \left (a \sqrt {x}+8\right )}}{\sqrt {x}\, a y-4 \sqrt {x \left (a \sqrt {x}+8\right )}}}\, \left (\sqrt {x}\, a y-4 \sqrt {x \left (a \sqrt {x}+8\right )}\right )} = 0
\]
Maple trace
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
differential order: 1; looking for linear symmetries
trying exact
trying Abel
<- Abel successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )=\frac {3 y \left (x \right )}{\sqrt {a \,x^{{3}/{2}}+8 x}}+1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\frac {3 y \left (x \right )}{\sqrt {a \,x^{{3}/{2}}+8 x}}+1}{y \left (x \right )} \end {array} \]
2.23.5.2 ✗ 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
2.23.5.3 ✗ 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