2.24.19 Problem 31
Internal
problem
ID
[13583]
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
:
31
Date
solved
:
Friday, December 19, 2025 at 07:30:50 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*}
y y^{\prime }-\frac {a \left (x +1\right ) y}{2 x^{{7}/{4}}}&=\frac {a^{2} \left (x -1\right ) \left (3 x +5\right )}{4 x^{{5}/{2}}} \\
\end{align*}
Unknown ode type.
2.24.19.1 ✓ Maple. Time used: 0.004 (sec). Leaf size: 187
ode:=y(x)*diff(y(x),x)-1/2*a*(x+1)/x^(7/4)*y(x) = 1/4*a^2*(x-1)*(5+3*x)/x^(5/2);
dsolve(ode,y(x), singsol=all);
\[
\frac {\frac {36 \sqrt {13}\, \sqrt {-\frac {\left (x -1\right ) a +x^{{3}/{4}} y}{x^{{3}/{4}} \left (y+x^{{1}/{4}} a \right )}}\, 55^{{1}/{6}} \left (x -\frac {15}{2}\right ) \left (\frac {\left (3 x +5\right ) a +3 x^{{3}/{4}} y}{x^{{3}/{4}} \left (y+x^{{1}/{4}} a \right )}\right )^{{5}/{6}}}{20449}+1458000 \left (\frac {a}{x^{{3}/{4}} \left (y+x^{{1}/{4}} a \right )}\right )^{{4}/{3}} x \left (\int _{}^{-\frac {90 \left (2 x^{{3}/{4}} y+2 a x -15 a \right )}{143 \left (x^{{3}/{4}} y+a x \right )}}\frac {\textit {\_a} \left (13 \textit {\_a} +90\right )^{{5}/{6}} \sqrt {11 \textit {\_a} -90}}{\left (143 \textit {\_a} +180\right )^{{4}/{3}} \left (20449 \textit {\_a}^{3}-1190700 \textit {\_a} -1458000\right )}d \textit {\_a} +\frac {c_1}{1458000}\right )}{x \left (\frac {a}{x^{{3}/{4}} \left (y+x^{{1}/{4}} a \right )}\right )^{{4}/{3}}} = 0
\]
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
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-\frac {a \left (x +1\right ) y \left (x \right )}{2 x^{{7}/{4}}}=\frac {a^{2} \left (x -1\right ) \left (3 x +5\right )}{4 x^{{5}/{2}}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\frac {a \left (x +1\right ) y \left (x \right )}{2 x^{{7}/{4}}}+\frac {a^{2} \left (x -1\right ) \left (3 x +5\right )}{4 x^{{5}/{2}}}}{y \left (x \right )} \end {array} \]
2.24.19.2 ✗ Mathematica
ode=y[x]*D[y[x],x]-1/2*a*(x+1)*x^(-7/4)*y[x]==1/4*a^2*(x-1)*(3*x+5)*x^(-5/2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
2.24.19.3 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a**2*(x - 1)*(3*x + 5)/(4*x**(5/2)) - a*(x + 1)*y(x)/(2*x**(7/4)) + y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out