2.24.36 Problem 52
Internal
problem
ID
[13600]
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
:
52
Date
solved
:
Friday, December 19, 2025 at 08:06:25 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*}
y y^{\prime }-\frac {a \left (2 x -1\right ) y}{x^{{5}/{2}}}&=\frac {a^{2} \left (x -1\right ) \left (3 x +1\right )}{2 x^{4}} \\
\end{align*}
Unknown ode type.
2.24.36.1 ✓ Maple. Time used: 0.007 (sec). Leaf size: 189
ode:=y(x)*diff(y(x),x)-a*(2*x-1)/x^(5/2)*y(x) = 1/2*a^2*(x-1)*(1+3*x)/x^4;
dsolve(ode,y(x), singsol=all);
\[
\frac {\frac {18 \sqrt {\frac {\left (x -1\right ) a +x^{{3}/{2}} y}{x \left (y \sqrt {x}+a \right )}}\, \sqrt {5}\, 7^{{5}/{6}} \left (x +\frac {3}{2}\right ) \left (\frac {\left (-3 x -1\right ) a -3 x^{{3}/{2}} y}{x \left (y \sqrt {x}+a \right )}\right )^{{1}/{6}}}{1225}+1458 x \left (-\frac {a}{x \left (y \sqrt {x}+a \right )}\right )^{{2}/{3}} \left (\int _{}^{\frac {-\frac {18 x^{{3}/{2}} y}{35}+\frac {9 \left (-2 x -3\right ) a}{35}}{x \left (y \sqrt {x}+a \right )}}\frac {\textit {\_a} \left (5 \textit {\_a} -9\right )^{{1}/{6}} \sqrt {7 \textit {\_a} +9}}{\left (35 \textit {\_a} +18\right )^{{2}/{3}} \left (1225 \textit {\_a}^{3}-3159 \textit {\_a} -1458\right )}d \textit {\_a} +\frac {c_1}{1458}\right )}{\left (-\frac {a}{x \left (y \sqrt {x}+a \right )}\right )^{{2}/{3}} x} = 0
\]
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
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-\frac {a \left (2 x -1\right ) y \left (x \right )}{x^{{5}/{2}}}=\frac {a^{2} \left (x -1\right ) \left (3 x +1\right )}{2 x^{4}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\frac {a \left (2 x -1\right ) y \left (x \right )}{x^{{5}/{2}}}+\frac {a^{2} \left (x -1\right ) \left (3 x +1\right )}{2 x^{4}}}{y \left (x \right )} \end {array} \]
2.24.36.2 ✗ Mathematica
ode=y[x]*D[y[x],x]-a*(2*x-1)*x^(-5/2)*y[x]==1/2*a^2*(x-1)*(3*x+1)*x^(-4);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
2.24.36.3 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a**2*(x - 1)*(3*x + 1)/(2*x**4) - a*(2*x - 1)*y(x)/x**(5/2) + y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out