2.26.3 Problem 3
Internal
problem
ID
[13639]
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
:
3
Date
solved
:
Friday, December 19, 2025 at 09:47:50 AM
CAS
classification
:
[_Abel]
\begin{align*}
y^{\prime }&=-y^{3}+\left (x a +b \right ) y^{2} \\
\end{align*}
Unknown ode type.
2.26.3.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 241
ode:=diff(y(x),x) = -y(x)^3+(a*x+b)*y(x)^2;
dsolve(ode,y(x), singsol=all);
\[
\frac {\left (-a^{2}\right )^{{1}/{3}} \left (a x +b \right ) \left (\operatorname {AiryBi}\left (-\frac {\left (y \left (a x +b \right )^{2}+2 a \right ) 2^{{2}/{3}}}{4 \left (-a^{2}\right )^{{1}/{3}} y}\right ) c_1 +\operatorname {AiryAi}\left (-\frac {\left (y \left (a x +b \right )^{2}+2 a \right ) 2^{{2}/{3}}}{4 \left (-a^{2}\right )^{{1}/{3}} y}\right )\right ) 2^{{1}/{3}}+2 \left (\operatorname {AiryBi}\left (1, -\frac {\left (y \left (a x +b \right )^{2}+2 a \right ) 2^{{2}/{3}}}{4 \left (-a^{2}\right )^{{1}/{3}} y}\right ) c_1 +\operatorname {AiryAi}\left (1, -\frac {\left (y \left (a x +b \right )^{2}+2 a \right ) 2^{{2}/{3}}}{4 \left (-a^{2}\right )^{{1}/{3}} y}\right )\right ) a}{2^{{1}/{3}} \left (a x +b \right ) \left (-a^{2}\right )^{{1}/{3}} \operatorname {AiryBi}\left (-\frac {\left (y \left (a x +b \right )^{2}+2 a \right ) 2^{{2}/{3}}}{4 \left (-a^{2}\right )^{{1}/{3}} y}\right )+2 \operatorname {AiryBi}\left (1, -\frac {\left (y \left (a x +b \right )^{2}+2 a \right ) 2^{{2}/{3}}}{4 \left (-a^{2}\right )^{{1}/{3}} y}\right ) a} = 0
\]
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
<- Abel successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=-y \left (x \right )^{3}+\left (a x +b \right ) y \left (x \right )^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=-y \left (x \right )^{3}+\left (a x +b \right ) y \left (x \right )^{2} \end {array} \]
2.26.3.2 ✓ Mathematica. Time used: 0.394 (sec). Leaf size: 223
ode=D[y[x],x]==-y[x]^3+(a*x+b)*y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {\operatorname {AiryAiPrime}\left (-\frac {\sqrt [3]{-\frac {1}{2}} (b+a x)^2}{2 a^{2/3}}-\frac {\sqrt [3]{-\frac {1}{2}} \sqrt [3]{a}}{y(x)}\right )-\frac {\left (-\frac {1}{2}\right )^{2/3} (a x+b) \operatorname {AiryAi}\left (-\frac {\sqrt [3]{-\frac {1}{2}} (b+a x)^2}{2 a^{2/3}}-\frac {\sqrt [3]{-\frac {1}{2}} \sqrt [3]{a}}{y(x)}\right )}{\sqrt [3]{a}}}{\operatorname {AiryBiPrime}\left (-\frac {\sqrt [3]{-\frac {1}{2}} (b+a x)^2}{2 a^{2/3}}-\frac {\sqrt [3]{-\frac {1}{2}} \sqrt [3]{a}}{y(x)}\right )-\frac {\left (-\frac {1}{2}\right )^{2/3} (a x+b) \operatorname {AiryBi}\left (-\frac {\sqrt [3]{-\frac {1}{2}} (b+a x)^2}{2 a^{2/3}}-\frac {\sqrt [3]{-\frac {1}{2}} \sqrt [3]{a}}{y(x)}\right )}{\sqrt [3]{a}}}+c_1=0,y(x)\right ]
\]
2.26.3.3 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq((-a*x - b)*y(x)**2 + y(x)**3 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
RecursionError : maximum recursion depth exceeded