2.26.17 Problem 17

Internal problem ID [13653]
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 : Friday, December 19, 2025 at 10:02:56 AM
CAS classification : [_rational, _Abel]

\begin{align*} y^{\prime } x^{2}&=y^{3}-3 y a^{2} x^{4}+2 a^{3} x^{6}+2 a \,x^{3} \\ \end{align*}

Unknown ode type.

2.26.17.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 276

ode:=x^2*diff(y(x),x) = y(x)^3-3*y(x)*a^2*x^4+2*a^3*x^6+2*a*x^3; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {a \left (18^{{2}/{3}} \left (a^{2}\right )^{{2}/{3}} \operatorname {RootOf}\left (\operatorname {AiryBi}\left (-\frac {-18 \textit {\_Z}^{2} \left (a^{2}\right )^{{1}/{3}} x +18^{{2}/{3}}}{18 \left (a^{2}\right )^{{1}/{3}} x}\right ) c_1 \textit {\_Z} +\textit {\_Z} \operatorname {AiryAi}\left (-\frac {-18 \textit {\_Z}^{2} \left (a^{2}\right )^{{1}/{3}} x +18^{{2}/{3}}}{18 \left (a^{2}\right )^{{1}/{3}} x}\right )+\operatorname {AiryBi}\left (1, -\frac {-18 \textit {\_Z}^{2} \left (a^{2}\right )^{{1}/{3}} x +18^{{2}/{3}}}{18 \left (a^{2}\right )^{{1}/{3}} x}\right ) c_1 +\operatorname {AiryAi}\left (1, -\frac {-18 \textit {\_Z}^{2} \left (a^{2}\right )^{{1}/{3}} x +18^{{2}/{3}}}{18 \left (a^{2}\right )^{{1}/{3}} x}\right )\right ) x^{2}-9 a^{2} x^{3}+3\right )}{-9 a^{2} x +\operatorname {RootOf}\left (\operatorname {AiryBi}\left (-\frac {-18 \textit {\_Z}^{2} \left (a^{2}\right )^{{1}/{3}} x +18^{{2}/{3}}}{18 \left (a^{2}\right )^{{1}/{3}} x}\right ) c_1 \textit {\_Z} +\textit {\_Z} \operatorname {AiryAi}\left (-\frac {-18 \textit {\_Z}^{2} \left (a^{2}\right )^{{1}/{3}} x +18^{{2}/{3}}}{18 \left (a^{2}\right )^{{1}/{3}} x}\right )+\operatorname {AiryBi}\left (1, -\frac {-18 \textit {\_Z}^{2} \left (a^{2}\right )^{{1}/{3}} x +18^{{2}/{3}}}{18 \left (a^{2}\right )^{{1}/{3}} x}\right ) c_1 +\operatorname {AiryAi}\left (1, -\frac {-18 \textit {\_Z}^{2} \left (a^{2}\right )^{{1}/{3}} x +18^{{2}/{3}}}{18 \left (a^{2}\right )^{{1}/{3}} x}\right )\right ) \left (a^{2}\right )^{{2}/{3}} 18^{{2}/{3}}} \]

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

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left (\frac {d}{d x}y \left (x \right )\right )=y \left (x \right )^{3}-3 a^{2} x^{4} y \left (x \right )+2 a^{3} x^{6}+2 a \,x^{3} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )^{3}-3 a^{2} x^{4} y \left (x \right )+2 a^{3} x^{6}+2 a \,x^{3}}{x^{2}} \end {array} \]

2.26.17.2 ✓ 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]
 

\[ \text {Solve}\left [\frac {\left (\left (-\frac {3}{2}\right )^{2/3} a^{2/3} x-\frac {1}{\sqrt [3]{-3} 2^{2/3} \sqrt [3]{a} y(x)-\sqrt [3]{-3} 2^{2/3} a^{4/3} x^2}\right ) \operatorname {AiryAi}\left (\left (\left (-\frac {3}{2}\right )^{2/3} a^{2/3} x-\frac {1}{\sqrt [3]{-3} 2^{2/3} \sqrt [3]{a} y(x)-\sqrt [3]{-3} 2^{2/3} a^{4/3} x^2}\right )^2+\frac {\sqrt [3]{-\frac {1}{2}}}{3^{2/3} a^{2/3} x}\right )+\operatorname {AiryAiPrime}\left (\left (\left (-\frac {3}{2}\right )^{2/3} a^{2/3} x-\frac {1}{\sqrt [3]{-3} 2^{2/3} \sqrt [3]{a} y(x)-\sqrt [3]{-3} 2^{2/3} a^{4/3} x^2}\right )^2+\frac {\sqrt [3]{-\frac {1}{2}}}{3^{2/3} a^{2/3} x}\right )}{\left (\left (-\frac {3}{2}\right )^{2/3} a^{2/3} x-\frac {1}{\sqrt [3]{-3} 2^{2/3} \sqrt [3]{a} y(x)-\sqrt [3]{-3} 2^{2/3} a^{4/3} x^2}\right ) \operatorname {AiryBi}\left (\left (\left (-\frac {3}{2}\right )^{2/3} a^{2/3} x-\frac {1}{\sqrt [3]{-3} 2^{2/3} \sqrt [3]{a} y(x)-\sqrt [3]{-3} 2^{2/3} a^{4/3} x^2}\right )^2+\frac {\sqrt [3]{-\frac {1}{2}}}{3^{2/3} a^{2/3} x}\right )+\operatorname {AiryBiPrime}\left (\left (\left (-\frac {3}{2}\right )^{2/3} a^{2/3} x-\frac {1}{\sqrt [3]{-3} 2^{2/3} \sqrt [3]{a} y(x)-\sqrt [3]{-3} 2^{2/3} a^{4/3} x^2}\right )^2+\frac {\sqrt [3]{-\frac {1}{2}}}{3^{2/3} a^{2/3} x}\right )}+c_1=0,y(x)\right ] \]

2.26.17.3 ✗ 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)
 

Timed Out
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0