55.2.26 problem 26
Internal
problem
ID
[13252]
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
:
26
Date
solved
:
Wednesday, October 01, 2025 at 04:22:45 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=y^{2}+a \,x^{n} y+b \,x^{n -1} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 267
ode:=diff(y(x),x) = y(x)^2+a*x^n*y(x)+b*x^(n-1);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-a \left (n +1\right ) \left (a -b \right ) \operatorname {KummerM}\left (\frac {\left (n +2\right ) a -b}{\left (n +1\right ) a}, \frac {n +2}{n +1}, \frac {a \,x^{n} x}{n +1}\right )+\left (c_1 \left (a -b \right ) \operatorname {KummerU}\left (\frac {2+n -\frac {b}{a}}{n +1}, \frac {n +2}{n +1}, \frac {a \,x^{n} x}{n +1}\right )-a \left (\operatorname {KummerU}\left (\frac {a -b}{\left (n +1\right ) a}, \frac {n +2}{n +1}, \frac {a \,x^{n} x}{n +1}\right ) c_1 +\operatorname {KummerM}\left (\frac {a -b}{\left (n +1\right ) a}, \frac {n +2}{n +1}, \frac {a \,x^{n} x}{n +1}\right )\right ) \left (n +1\right )\right ) b}{\left (n +1\right ) a^{2} x \left (\operatorname {KummerU}\left (\frac {a -b}{\left (n +1\right ) a}, \frac {n +2}{n +1}, \frac {a \,x^{n} x}{n +1}\right ) c_1 +\operatorname {KummerM}\left (\frac {a -b}{\left (n +1\right ) a}, \frac {n +2}{n +1}, \frac {a \,x^{n} x}{n +1}\right )\right )}
\]
✓ Mathematica. Time used: 0.297 (sec). Leaf size: 453
ode=D[y[x],x]==y[x]^2+a*x^n*y[x]+b*x^(n-1);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\left (x^n\right )^{\frac {1}{n}} \left (-(-1)^{\frac {1}{n+1}} n (n+2) a^{\frac {1}{n+1}} \operatorname {Hypergeometric1F1}\left (\frac {a-b}{n a+a},\frac {n+2}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )+x^n \left (-(-1)^{\frac {1}{n+1}} n (a-b) a^{\frac {1}{n+1}} \left (x^n\right )^{\frac {1}{n}} \operatorname {Hypergeometric1F1}\left (\frac {n a+2 a-b}{n a+a},\frac {2 n+3}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )+b c_1 \left (\frac {1}{n}+1\right )^{\frac {1}{n+1}} n^{\frac {1}{n+1}} (n+2) \operatorname {Hypergeometric1F1}\left (\frac {n a+a-b}{n a+a},\frac {2 n+1}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )\right )\right )}{n (n+2) x \left ((-1)^{\frac {1}{n+1}} a^{\frac {1}{n+1}} \left (x^n\right )^{\frac {1}{n}} \operatorname {Hypergeometric1F1}\left (\frac {a-b}{n a+a},\frac {n+2}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )+c_1 \left (\frac {1}{n}+1\right )^{\frac {1}{n+1}} n^{\frac {1}{n+1}} \operatorname {Hypergeometric1F1}\left (-\frac {b}{n a+a},\frac {n}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )\right )}\\ y(x)&\to \frac {b x^{n-1} \left (x^n\right )^{\frac {1}{n}} \operatorname {Hypergeometric1F1}\left (\frac {n a+a-b}{n a+a},\frac {2 n+1}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )}{n \operatorname {Hypergeometric1F1}\left (-\frac {b}{n a+a},\frac {n}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )} \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*y(x) - 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**n*y(x) - b*x**(n - 1) - y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method