61.2.66 problem 66
Internal
problem
ID
[11993]
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
:
Sunday, March 30, 2025 at 10:05:45 PM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} x^{3} y^{\prime }&=a \,x^{3} 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: 1.523 (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