55.26.6 problem 6

Internal problem ID [13753]
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 : 6
Date solved : Thursday, October 02, 2025 at 07:57:59 AM
CAS classification : [_Abel]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 208

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

\begin{align*} y &= -b x \\ y &= -\frac {2 \,{\mathrm e}^{-a \,x^{3} b^{2}}}{\sqrt {\frac {-8 \,{\mathrm e}^{-2 a \,x^{3} b^{2}} \left (a \,x^{3} b^{2}\right )^{{1}/{6}} a x +4 \left (a \,x^{3} b^{2}\right )^{{1}/{6}} c_1 -3 \,{\mathrm e}^{-a \,x^{3} b^{2}} a x 2^{{5}/{6}} \operatorname {WhittakerM}\left (\frac {1}{6}, \frac {2}{3}, 2 a \,x^{3} b^{2}\right )}{\left (a \,x^{3} b^{2}\right )^{{1}/{6}}}}}-b x \\ y &= \frac {2 \,{\mathrm e}^{-a \,x^{3} b^{2}}}{\sqrt {\frac {-8 \,{\mathrm e}^{-2 a \,x^{3} b^{2}} \left (a \,x^{3} b^{2}\right )^{{1}/{6}} a x +4 \left (a \,x^{3} b^{2}\right )^{{1}/{6}} c_1 -3 \,{\mathrm e}^{-a \,x^{3} b^{2}} a x 2^{{5}/{6}} \operatorname {WhittakerM}\left (\frac {1}{6}, \frac {2}{3}, 2 a \,x^{3} b^{2}\right )}{\left (a \,x^{3} b^{2}\right )^{{1}/{6}}}}}-b x \\ \end{align*}

✓ Mathematica. Time used: 4.472 (sec). Leaf size: 138

ode=D[y[x],x]==a*y[x]^3+3*a*b*x*y[x]^2-b-2*a*b^3*x^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -b x-\frac {e^{-a b^2 x^3}}{\sqrt {\frac {2^{2/3} a x \Gamma \left (\frac {1}{3},2 a b^2 x^3\right )}{3 \sqrt [3]{a b^2 x^3}}+c_1}}\\ y(x)&\to -b x+\frac {e^{-a b^2 x^3}}{\sqrt {\frac {2^{2/3} a x \Gamma \left (\frac {1}{3},2 a b^2 x^3\right )}{3 \sqrt [3]{a b^2 x^3}}+c_1}}\\ y(x)&\to -b x \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(2*a*b**3*x**3 - 3*a*b*x*y(x)**2 - a*y(x)**3 + b + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

RecursionError : maximum recursion depth exceeded