2.24.35 Problem 51
Internal
problem
ID
[13599]
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
1.3.3-2.
Problem
number
:
51
Date
solved
:
Friday, December 19, 2025 at 08:04:05 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*}
y y^{\prime }-\frac {a \left (x +4\right ) y}{5 x^{{8}/{5}}}&=\frac {a^{2} \left (x -1\right ) \left (3 x +7\right )}{5 x^{{11}/{5}}} \\
\end{align*}
Unknown ode type.
2.24.35.1 ✓ Maple. Time used: 0.008 (sec). Leaf size: 190
ode:=y(x)*diff(y(x),x)-1/5*a*(x+4)/x^(8/5)*y(x) = 1/5*a^2*(x-1)*(7+3*x)/x^(11/5);
dsolve(ode,y(x), singsol=all);
\[
\frac {\frac {360 \,2^{{1}/{3}} \sqrt {-\frac {\left (x -1\right ) a +y x^{{3}/{5}}}{x^{{3}/{5}} \left (y+x^{{2}/{5}} a \right )}}\, \sqrt {17}\, 91^{{5}/{6}} \left (x -\frac {21}{4}\right ) \left (\frac {\left (3 x +7\right ) a +3 y x^{{3}/{5}}}{x^{{3}/{5}} \left (y+x^{{2}/{5}} a \right )}\right )^{{7}/{6}}}{4444531}+31255875 x \left (\int _{}^{-\frac {315 \left (4 y x^{{3}/{5}}+4 a x -21 a \right )}{884 \left (y x^{{3}/{5}}+a x \right )}}\frac {\sqrt {52 \textit {\_a} -315}\, \left (68 \textit {\_a} +315\right )^{{1}/{6}} \textit {\_a}}{\left (11492 \textit {\_a}^{2}-53235 \textit {\_a} -99225\right ) \left (221 \textit {\_a} +315\right )^{{5}/{3}}}d \textit {\_a} +\frac {c_1}{31255875}\right ) \left (\frac {a}{x^{{3}/{5}} \left (y+x^{{2}/{5}} a \right )}\right )^{{5}/{3}}}{\left (\frac {a}{x^{{3}/{5}} \left (y+x^{{2}/{5}} a \right )}\right )^{{5}/{3}} x} = 0
\]
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
Looking for potential symmetries
Looking for potential symmetries
<- Abel successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-\frac {a \left (x +4\right ) y \left (x \right )}{5 x^{{8}/{5}}}=\frac {a^{2} \left (x -1\right ) \left (3 x +7\right )}{5 x^{{11}/{5}}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\frac {a \left (x +4\right ) y \left (x \right )}{5 x^{{8}/{5}}}+\frac {a^{2} \left (x -1\right ) \left (3 x +7\right )}{5 x^{{11}/{5}}}}{y \left (x \right )} \end {array} \]
2.24.35.2 ✗ Mathematica
ode=y[x]*D[y[x],x]-1/5*a*(x+4)*x^(-8/5)*y[x]==1/5*a^2*(x-1)*(3*x+7)*x^(-11/5);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Timed out
2.24.35.3 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a**2*(x - 1)*(3*x + 7)/(5*x**(11/5)) - a*(x + 4)*y(x)/(5*x**(8/5)) + y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out
Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0]
Sympy version 1.14.0