2.26.4 Problem 4
Internal
problem
ID
[13640]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.4.
Equations
Containing
Polynomial
Functions
of
y.
subsection
1.4.1-2
Abel
equations
of
the
first
kind.
Problem
number
:
4
Date
solved
:
Friday, December 19, 2025 at 09:48:55 AM
CAS
classification
:
[_rational, _Abel]
\begin{align*}
y^{\prime }&=-y^{3}+\frac {y^{2}}{\left (x a +b \right )^{2}} \\
\end{align*}
Unknown ode type.
2.26.4.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 181
ode:=diff(y(x),x) = -y(x)^3+1/(a*x+b)^2*y(x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {2^{{1}/{3}} a^{2} \left (a x +b \right )}{2 \left (a^{2}\right )^{{2}/{3}} \left (a x +b \right ) \operatorname {RootOf}\left (\operatorname {AiryBi}\left (-\frac {a^{2} 2^{{2}/{3}} x +a 2^{{2}/{3}} b -2 \textit {\_Z}^{2} \left (a^{2}\right )^{{1}/{3}}}{2 \left (a^{2}\right )^{{1}/{3}}}\right ) c_1 \textit {\_Z} +\operatorname {AiryBi}\left (1, -\frac {a^{2} 2^{{2}/{3}} x +a 2^{{2}/{3}} b -2 \textit {\_Z}^{2} \left (a^{2}\right )^{{1}/{3}}}{2 \left (a^{2}\right )^{{1}/{3}}}\right ) c_1 +\textit {\_Z} \operatorname {AiryAi}\left (-\frac {a^{2} 2^{{2}/{3}} x +a 2^{{2}/{3}} b -2 \textit {\_Z}^{2} \left (a^{2}\right )^{{1}/{3}}}{2 \left (a^{2}\right )^{{1}/{3}}}\right )+\operatorname {AiryAi}\left (1, -\frac {a^{2} 2^{{2}/{3}} x +a 2^{{2}/{3}} b -2 \textit {\_Z}^{2} \left (a^{2}\right )^{{1}/{3}}}{2 \left (a^{2}\right )^{{1}/{3}}}\right )\right )-2^{{1}/{3}} a}
\]
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
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=-y \left (x \right )^{3}+\frac {y \left (x \right )^{2}}{\left (a x +b \right )^{2}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=-y \left (x \right )^{3}+\frac {y \left (x \right )^{2}}{\left (a x +b \right )^{2}} \end {array} \]
2.26.4.2 ✓ Mathematica. Time used: 0.558 (sec). Leaf size: 1041
ode=D[y[x],x]==-y[x]^3+(a*x+b)^(-2)*y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} \text {Solution too large to show}\end{align*}
2.26.4.3 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(y(x)**3 + Derivative(y(x), x) - y(x)**2/(a*x + b)**2,0)
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