61.24.2 problem 2

Internal problem ID [12336]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.3-2.
Problem number : 2
Date solved : Monday, March 31, 2025 at 05:11:02 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 826

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

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 0.217 (sec). Leaf size: 145

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

\[ \text {Solve}\left [\int _1^x\frac {\left (-2 b^3-9 c b\right )^{2/3}}{9 K[2] (a K[2] (b+a K[2])-c)}dK[2]+c_1=\int _1^{\frac {3 x (a x (b+a x)-c)-(b+3 a x) y(x)}{\sqrt [3]{-2 b^3-9 c b} y(x)}}\frac {1}{K[1]^3+\frac {3 \sqrt [3]{-1} \left (b^2+3 c\right ) K[1]}{b^{2/3} \left (2 b^2+9 c\right )^{2/3}}+1}dK[1],y(x)\right ] \]

✗ Sympy

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

Timed Out