55.2.22 problem 22
Internal
problem
ID
[13248]
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
:
22
Date
solved
:
Wednesday, October 01, 2025 at 04:12:55 AM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} \left (a \,x^{n}+b \right ) y^{\prime }&=b y^{2}+a \,x^{n -2} \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 224
ode:=(a*x^n+b)*diff(y(x),x) = b*y(x)^2+a*x^(n-2);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (\frac {a \,x^{n}+b}{b}\right )^{\frac {2}{n}} \left (a n c_1 \left (a^{2} x^{3 n}+2 a b \,x^{2 n}+b^{2} x^{n}\right ) \operatorname {hypergeom}\left (\left [2, \frac {1+n}{n}\right ], \left [\frac {-1+2 n}{n}\right ], -\frac {a \,x^{n}}{b}\right )-\left (a c_1 \left (a \,x^{2 n}+x^{n} b \right ) \operatorname {hypergeom}\left (\left [1, \frac {1}{n}\right ], \left [\frac {-1+n}{n}\right ], -\frac {a \,x^{n}}{b}\right )+\left (\frac {a \,x^{n}+b}{b}\right )^{-\frac {2}{n}} b \left (x^{1+n} a +b x \right )\right ) \left (-1+n \right ) b \right )}{b^{2} \left (-1+n \right ) x \left (a \,x^{n}+b \right ) \left (x +\operatorname {hypergeom}\left (\left [1, \frac {1}{n}\right ], \left [\frac {-1+n}{n}\right ], -\frac {a \,x^{n}}{b}\right ) c_1 \left (\frac {a \,x^{n}+b}{b}\right )^{\frac {2}{n}}\right )}
\]
✓ Mathematica. Time used: 1.201 (sec). Leaf size: 289
ode=(a*x^n+b)*D[y[x],x]==b*y[x]^2+a*x^(n-2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {-b^2 (-1)^{\frac {1}{n}} (n-1) \left (-\frac {a x^n}{b}\right )^{\frac {1}{n}}-a b c_1 (n-1) x^n \left (\frac {a x^n}{b}+1\right )^{2/n} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},\frac {n-1}{n},-\frac {a x^n}{b}\right )+a c_1 n x^n \left (a x^n+b\right ) \left (\frac {a x^n}{b}+1\right )^{2/n} \operatorname {Hypergeometric2F1}\left (2,1+\frac {1}{n},2-\frac {1}{n},-\frac {a x^n}{b}\right )}{b^2 (n-1) x \left ((-1)^{\frac {1}{n}} \left (-\frac {a x^n}{b}\right )^{\frac {1}{n}}+c_1 \left (\frac {a x^n}{b}+1\right )^{2/n} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},\frac {n-1}{n},-\frac {a x^n}{b}\right )\right )}\\ y(x)&\to \frac {a x^{n-1} \left (\frac {n \left (a x^n+b\right ) \operatorname {Hypergeometric2F1}\left (2,1+\frac {1}{n},2-\frac {1}{n},-\frac {a x^n}{b}\right )}{\operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},\frac {n-1}{n},-\frac {a x^n}{b}\right )}+b (-n)+b\right )}{b^2 (n-1)} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
n = symbols("n")
y = Function("y")
ode = Eq(-a*x**(n - 2) - b*y(x)**2 + (a*x**n + b)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (a*x**(n - 2) + b*y(x)**2)/(a*x**n + b) cannot be solved by the factorable group method