2.22.18 Problem 22
Internal
problem
ID
[13513]
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
1.3.1-2.
Solvable
equations
and
their
solutions
Problem
number
:
22
Date
solved
:
Friday, December 19, 2025 at 05:21:54 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*}
y y^{\prime }-y&=-\frac {4 x}{25}+\frac {A}{\sqrt {x}} \\
\end{align*}
Unknown ode type.
2.22.18.1 ✓ Maple. Time used: 0.001 (sec). Leaf size: 189
ode:=y(x)*diff(y(x),x)-y(x) = -4/25*x+A/x^(1/2);
dsolve(ode,y(x), singsol=all);
\[
c_1 +\frac {31250 \left (-\frac {y^{5}}{10}+A^{2} y^{2}\right ) x^{{3}/{2}}+62500 \left (\frac {y^{4}}{5}-2 A^{2} y\right ) x^{{5}/{2}}+20000 \left (-y^{3}+4 A^{2}\right ) x^{{7}/{2}}-16000 A \,x^{5}+60000 A y x^{4}-75000 A y^{2} x^{3}+31250 A y^{3} x^{2}-125000 A^{3} x^{2}+16000 x^{{9}/{2}} y^{2}-6400 x^{{11}/{2}} y+1024 x^{{13}/{2}}}{x^{{3}/{2}} \sqrt {A \,x^{{3}/{2}}}\, A \left (25 A \sqrt {x}-4 \left (x -\frac {5 y}{4}\right )^{2}\right ) \left (\frac {100 A \sqrt {x}-16 x^{2}+40 x y-25 y^{2}}{A \sqrt {x}}\right )^{{3}/{2}}} = 0
\]
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
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=-\frac {4 x}{25}+\frac {A}{\sqrt {x}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )-\frac {4 x}{25}+\frac {A}{\sqrt {x}}}{y \left (x \right )} \end {array} \]
2.22.18.2 ✗ Mathematica
ode=y[x]*D[y[x],x]-y[x]==-4/25*x+A*x^(-1/2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
2.22.18.3 ✗ Sympy
from sympy import *
x = symbols("x")
A = symbols("A")
y = Function("y")
ode = Eq(-A/sqrt(x) + 4*x/25 + y(x)*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out