55.2.65 problem 65

Internal problem ID [13291]
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 : 65
Date solved : Wednesday, October 01, 2025 at 06:00:50 AM
CAS classification : [_rational, _Riccati]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 435

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

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

✓ Mathematica. Time used: 1.281 (sec). Leaf size: 907

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

NotImplementedError : The given ODE -a*y(x)**2 - b*y(x)/x - c*y(x)/x**3 - s/x**2 + Derivative(y(x), x) cannot be solved by the factorable group method