61.2.65 problem 65
Internal
problem
ID
[11992]
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
:
Sunday, March 30, 2025 at 10:05:25 PM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} x^{3} y^{\prime }&=a \,x^{3} 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: 2.05 (sec). Leaf size: 1140
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