55.26.4 problem 4

Internal problem ID [13751]
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 : Sunday, October 12, 2025 at 05:28:13 AM
CAS classification : [_rational, _Abel]

\begin{align*} y^{\prime }&=-y^{3}+\frac {y^{2}}{\left (a x +b \right )^{2}} \end{align*}

✓ 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);
 

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

✓ 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*}

✗ 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