55.2.66 problem 66

Internal problem ID [13292]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number : 66
Date solved : Wednesday, October 01, 2025 at 06:02:07 AM
CAS classification : [_rational, _Riccati]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 438

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

\[ y = \frac {\left (\left (a \alpha +b +2\right ) c^{2}-a \beta \left (b +3\right ) c +a^{2} \beta ^{2}\right ) x c_1 \operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )-\left (\left (c^{2}+x \left (b +2\right ) c -a \beta x \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )-\left (-c^{2}-x \left (b +2\right ) c +a \beta x \right ) c_1 \operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )+\operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right ) \left (-\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c}{2}+\frac {\left (-b -3\right ) c}{2}+a \beta \right ) x \right ) c}{x^{2} c^{2} a \left (\operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right ) c_1 +\operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 a \beta }{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )\right )} \]

✓ Mathematica. Time used: 0.971 (sec). Leaf size: 1168

ode=x^3*D[y[x],x]==a*x^3*y[x]^2+x*(b*x+c)*y[x]+\[Alpha]*x+\[Beta]; 
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") 
Alpha = symbols("Alpha") 
BETA = symbols("BETA") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(-Alpha*x - BETA - a*x**3*y(x)**2 + x**3*Derivative(y(x), x) - x*(b*x + c)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out