55.26.17 problem 17
Internal
problem
ID
[13764]
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
:
17
Date
solved
:
Sunday, October 12, 2025 at 05:28:20 AM
CAS
classification
:
[_rational, _Abel]
\begin{align*} x^{2} y^{\prime }&=y^{3}-3 a^{2} x^{4} y+2 a^{3} x^{6}+2 a \,x^{3} \end{align*}
✓ Maple. Time used: 0.010 (sec). Leaf size: 276
ode:=x^2*diff(y(x),x) = y(x)^3-3*a^2*x^4*y(x)+2*a^3*x^6+2*a*x^3;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {a \left (18^{{2}/{3}} \left (a^{2}\right )^{{2}/{3}} \operatorname {RootOf}\left (\operatorname {AiryBi}\left (-\frac {-18 \textit {\_Z}^{2} \left (a^{2}\right )^{{1}/{3}} x +18^{{2}/{3}}}{18 \left (a^{2}\right )^{{1}/{3}} x}\right ) c_1 \textit {\_Z} +\textit {\_Z} \operatorname {AiryAi}\left (-\frac {-18 \textit {\_Z}^{2} \left (a^{2}\right )^{{1}/{3}} x +18^{{2}/{3}}}{18 \left (a^{2}\right )^{{1}/{3}} x}\right )+\operatorname {AiryBi}\left (1, -\frac {-18 \textit {\_Z}^{2} \left (a^{2}\right )^{{1}/{3}} x +18^{{2}/{3}}}{18 \left (a^{2}\right )^{{1}/{3}} x}\right ) c_1 +\operatorname {AiryAi}\left (1, -\frac {-18 \textit {\_Z}^{2} \left (a^{2}\right )^{{1}/{3}} x +18^{{2}/{3}}}{18 \left (a^{2}\right )^{{1}/{3}} x}\right )\right ) x^{2}-9 a^{2} x^{3}+3\right )}{-9 a^{2} x +\operatorname {RootOf}\left (\operatorname {AiryBi}\left (-\frac {-18 \textit {\_Z}^{2} \left (a^{2}\right )^{{1}/{3}} x +18^{{2}/{3}}}{18 \left (a^{2}\right )^{{1}/{3}} x}\right ) c_1 \textit {\_Z} +\textit {\_Z} \operatorname {AiryAi}\left (-\frac {-18 \textit {\_Z}^{2} \left (a^{2}\right )^{{1}/{3}} x +18^{{2}/{3}}}{18 \left (a^{2}\right )^{{1}/{3}} x}\right )+\operatorname {AiryBi}\left (1, -\frac {-18 \textit {\_Z}^{2} \left (a^{2}\right )^{{1}/{3}} x +18^{{2}/{3}}}{18 \left (a^{2}\right )^{{1}/{3}} x}\right ) c_1 +\operatorname {AiryAi}\left (1, -\frac {-18 \textit {\_Z}^{2} \left (a^{2}\right )^{{1}/{3}} x +18^{{2}/{3}}}{18 \left (a^{2}\right )^{{1}/{3}} x}\right )\right ) \left (a^{2}\right )^{{2}/{3}} 18^{{2}/{3}}}
\]
✓ Mathematica. Time used: 0.687 (sec). Leaf size: 463
ode=x^2*D[y[x],x]==y[x]^3-3*a^2*x^4*y[x]+2*a^3*x^6+2*a*x^3;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {\left (\left (-\frac {3}{2}\right )^{2/3} a^{2/3} x-\frac {1}{\sqrt [3]{-3} 2^{2/3} \sqrt [3]{a} y(x)-\sqrt [3]{-3} 2^{2/3} a^{4/3} x^2}\right ) \operatorname {AiryAi}\left (\left (\left (-\frac {3}{2}\right )^{2/3} a^{2/3} x-\frac {1}{\sqrt [3]{-3} 2^{2/3} \sqrt [3]{a} y(x)-\sqrt [3]{-3} 2^{2/3} a^{4/3} x^2}\right )^2+\frac {\sqrt [3]{-\frac {1}{2}}}{3^{2/3} a^{2/3} x}\right )+\operatorname {AiryAiPrime}\left (\left (\left (-\frac {3}{2}\right )^{2/3} a^{2/3} x-\frac {1}{\sqrt [3]{-3} 2^{2/3} \sqrt [3]{a} y(x)-\sqrt [3]{-3} 2^{2/3} a^{4/3} x^2}\right )^2+\frac {\sqrt [3]{-\frac {1}{2}}}{3^{2/3} a^{2/3} x}\right )}{\left (\left (-\frac {3}{2}\right )^{2/3} a^{2/3} x-\frac {1}{\sqrt [3]{-3} 2^{2/3} \sqrt [3]{a} y(x)-\sqrt [3]{-3} 2^{2/3} a^{4/3} x^2}\right ) \operatorname {AiryBi}\left (\left (\left (-\frac {3}{2}\right )^{2/3} a^{2/3} x-\frac {1}{\sqrt [3]{-3} 2^{2/3} \sqrt [3]{a} y(x)-\sqrt [3]{-3} 2^{2/3} a^{4/3} x^2}\right )^2+\frac {\sqrt [3]{-\frac {1}{2}}}{3^{2/3} a^{2/3} x}\right )+\operatorname {AiryBiPrime}\left (\left (\left (-\frac {3}{2}\right )^{2/3} a^{2/3} x-\frac {1}{\sqrt [3]{-3} 2^{2/3} \sqrt [3]{a} y(x)-\sqrt [3]{-3} 2^{2/3} a^{4/3} x^2}\right )^2+\frac {\sqrt [3]{-\frac {1}{2}}}{3^{2/3} a^{2/3} x}\right )}+c_1=0,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-2*a**3*x**6 + 3*a**2*x**4*y(x) - 2*a*x**3 + x**2*Derivative(y(x), x) - y(x)**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
RecursionError : maximum recursion depth exceeded