2.26.2 Problem 2
Internal
problem
ID
[13638]
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
:
2
Date
solved
:
Friday, December 19, 2025 at 09:46:57 AM
CAS
classification
:
[_Abel]
\begin{align*}
y^{\prime }&=-y^{3}+3 a^{2} x^{2} y-2 x^{3} a^{3}+a \\
\end{align*}
Unknown ode type.
2.26.2.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 136
ode:=diff(y(x),x) = -y(x)^3+3*a^2*x^2*y(x)-2*a^3*x^3+a;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {a \left (9 x^{3} a^{2}+2 \,18^{{1}/{3}} \left (-a^{2}\right )^{{1}/{3}} \operatorname {RootOf}\left (\operatorname {AiryBi}\left (\textit {\_Z} \right ) 18^{{1}/{3}} \left (-a^{2}\right )^{{1}/{3}} c_1 x +18^{{1}/{3}} \left (-a^{2}\right )^{{1}/{3}} x \operatorname {AiryAi}\left (\textit {\_Z} \right )+2 \operatorname {AiryBi}\left (1, \textit {\_Z}\right ) c_1 +2 \operatorname {AiryAi}\left (1, \textit {\_Z}\right )\right ) x +6\right )}{9 a^{2} x^{2}+2 \,18^{{1}/{3}} \left (-a^{2}\right )^{{1}/{3}} \operatorname {RootOf}\left (\operatorname {AiryBi}\left (\textit {\_Z} \right ) 18^{{1}/{3}} \left (-a^{2}\right )^{{1}/{3}} c_1 x +18^{{1}/{3}} \left (-a^{2}\right )^{{1}/{3}} x \operatorname {AiryAi}\left (\textit {\_Z} \right )+2 \operatorname {AiryBi}\left (1, \textit {\_Z}\right ) c_1 +2 \operatorname {AiryAi}\left (1, \textit {\_Z}\right )\right )}
\]
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
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=-y \left (x \right )^{3}+3 a^{2} x^{2} y \left (x \right )-2 a^{3} x^{3}+a \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=-y \left (x \right )^{3}+3 a^{2} x^{2} y \left (x \right )-2 a^{3} x^{3}+a \end {array} \]
2.26.2.2 ✓ Mathematica. Time used: 0.411 (sec). Leaf size: 307
ode=D[y[x],x]==-y[x]^3+3*a^2*x^2*y[x]-2*a^3*x^3+a;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {\operatorname {AiryAiPrime}\left (-\frac {3}{2} \sqrt [3]{-\frac {3}{2}} a^{4/3} x^2-\frac {1}{\frac {(-1)^{2/3} \sqrt [3]{\frac {2}{3}} y(x)}{\sqrt [3]{a}}-(-1)^{2/3} \sqrt [3]{\frac {2}{3}} a^{2/3} x}\right )-\left (-\frac {3}{2}\right )^{2/3} a^{2/3} x \operatorname {AiryAi}\left (-\frac {3}{2} \sqrt [3]{-\frac {3}{2}} a^{4/3} x^2-\frac {1}{\frac {(-1)^{2/3} \sqrt [3]{\frac {2}{3}} y(x)}{\sqrt [3]{a}}-(-1)^{2/3} \sqrt [3]{\frac {2}{3}} a^{2/3} x}\right )}{\operatorname {AiryBiPrime}\left (-\frac {3}{2} \sqrt [3]{-\frac {3}{2}} a^{4/3} x^2-\frac {1}{\frac {(-1)^{2/3} \sqrt [3]{\frac {2}{3}} y(x)}{\sqrt [3]{a}}-(-1)^{2/3} \sqrt [3]{\frac {2}{3}} a^{2/3} x}\right )-\left (-\frac {3}{2}\right )^{2/3} a^{2/3} x \operatorname {AiryBi}\left (-\frac {3}{2} \sqrt [3]{-\frac {3}{2}} a^{4/3} x^2-\frac {1}{\frac {(-1)^{2/3} \sqrt [3]{\frac {2}{3}} y(x)}{\sqrt [3]{a}}-(-1)^{2/3} \sqrt [3]{\frac {2}{3}} a^{2/3} x}\right )}+c_1=0,y(x)\right ]
\]
2.26.2.3 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(2*a**3*x**3 - 3*a**2*x**2*y(x) - a + y(x)**3 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE 2*a**3*x**3 - 3*a**2*x**2*d(x) - a + d(x)**3 + Derivative(d(x), x) cannot be solved by the factorable group method