55.2.11 problem 11
Internal
problem
ID
[13237]
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
:
11
Date
solved
:
Wednesday, October 01, 2025 at 03:48:56 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=\left (a \,x^{2 n}+b \,x^{n -1}\right ) y^{2}+c \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 1155
ode:=diff(y(x),x) = (a*x^(2*n)+b*x^(n-1))*y(x)^2+c;
dsolve(ode,y(x), singsol=all);
\[
\text {Expression too large to display}
\]
✓ Mathematica. Time used: 0.717 (sec). Leaf size: 836
ode=D[y[x],x]==(a*x^(2*n)+b*x^(n-1))*y[x]^2+c;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\sqrt {c} (n+1)^2 x^{-n} \left (L_{-\frac {\sqrt {c} b}{2 \sqrt {a} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )}{\sqrt {a} c_1 (n+1) \sqrt {-(n+1)^2} \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_1 \left (\sqrt {a} \sqrt {-(n+1)^2} n+b \sqrt {c} (n+1)\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {3 n+2}{n+1}\right ),\frac {n}{n+1}+1,\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+\sqrt {a} (n+1) \sqrt {-(n+1)^2} \left (L_{-\frac {\sqrt {c} b}{2 \sqrt {a} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+2 L_{-\frac {\sqrt {c} b}{2 \sqrt {a} \sqrt {-(n+1)^2}}-\frac {3 n+2}{2 n+2}}^{\frac {n}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )}\\ y(x)&\to \frac {\sqrt {c} (n+1)^2 x^{-n} \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )}{\sqrt {a} (n+1) \sqrt {-(n+1)^2} \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+\left (\sqrt {a} \sqrt {-(n+1)^2} n+b \sqrt {c} (n+1)\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}+2\right ),\frac {n}{n+1}+1,\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
n = symbols("n")
y = Function("y")
ode = Eq(-c - (a*x**(2*n) + b*x**(n - 1))*y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -a*x**(2*n)*y(x)**2 - b*x**(n - 1)*y(x)**2 - c + Derivative(y(x), x) cannot be solved by the factorable group method