55.2.5 problem 5
Internal
problem
ID
[13231]
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
:
5
Date
solved
:
Wednesday, October 01, 2025 at 03:41:56 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=y^{2}+a n \,x^{n -1}-a^{2} x^{2 n} \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 366
ode:=diff(y(x),x) = y(x)^2+a*n*x^(n-1)-a^2*x^(2*n);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-3 \left (n +2\right ) c_1 \left (\left (\frac {1}{3} n^{2}+n +\frac {2}{3}\right ) x^{-\frac {3 n}{2}}+a x \,x^{-\frac {n}{2}} \left (n +\frac {4}{3}\right )\right ) \operatorname {WhittakerM}\left (\frac {n +2}{2 n +2}, \frac {3+2 n}{2 n +2}, -\frac {2 a \,x^{n} x}{n +1}\right )+2 c_1 \left (n +1\right ) \left (\left (-\frac {1}{2} n^{2}-\frac {3}{2} n -1\right ) x^{-\frac {3 n}{2}}+x \left (\left (-1-\frac {n}{2}\right ) x^{-\frac {n}{2}}+a x \,x^{\frac {n}{2}}\right ) a \right ) \operatorname {WhittakerM}\left (-\frac {n}{2 n +2}, \frac {3+2 n}{2 n +2}, -\frac {2 a \,x^{n} x}{n +1}\right )+2 \left (n +2\right )^{2} c_1 \,x^{-\frac {3 n}{2}} {\mathrm e}^{\frac {a \,x^{n} x}{n +1}} \left (n +\frac {3}{2}\right ) \left (-\frac {2 a \,x^{n} x}{n +1}\right )^{\frac {3 n +4}{2 n +2}}+2 a \,x^{n} x^{2} {\mathrm e}^{-\frac {a \,x^{n} x}{n +1}}}{2 \left (-\frac {x^{-\frac {3 n}{2}} c_1 \left (n +2\right )^{2} \operatorname {WhittakerM}\left (\frac {n +2}{2 n +2}, \frac {3+2 n}{2 n +2}, -\frac {2 a \,x^{n} x}{n +1}\right )}{2}+c_1 \left (n +1\right ) \left (\left (-1-\frac {n}{2}\right ) x^{-\frac {3 n}{2}}+x^{-\frac {n}{2}} a x \right ) \operatorname {WhittakerM}\left (-\frac {n}{2 n +2}, \frac {3+2 n}{2 n +2}, -\frac {2 a \,x^{n} x}{n +1}\right )+{\mathrm e}^{-\frac {a \,x^{n} x}{n +1}} x \right ) x}
\]
✓ Mathematica. Time used: 0.686 (sec). Leaf size: 227
ode=D[y[x],x]==y[x]^2+a*n*x^(n-1)-a^2*x^(2*n);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {2^{\frac {1}{n+1}} (n+1) \left (-\frac {a x^{n+1}}{n+1}\right )^{\frac {1}{n+1}} \left (a x^n-c_1 e^{\frac {2 a x^{n+1}}{n+1}}\right )-a c_1 x^{n+1} \Gamma \left (\frac {1}{n+1},-\frac {2 a x^{n+1}}{n+1}\right )}{2^{\frac {1}{n+1}} (n+1) \left (-\frac {a x^{n+1}}{n+1}\right )^{\frac {1}{n+1}}-c_1 x \Gamma \left (\frac {1}{n+1},-\frac {2 a x^{n+1}}{n+1}\right )}\\ y(x)&\to \frac {2^{\frac {1}{n+1}} (n+1) e^{\frac {2 a x^{n+1}}{n+1}} \left (-\frac {a x^{n+1}}{n+1}\right )^{\frac {1}{n+1}}}{x \Gamma \left (\frac {1}{n+1},-\frac {2 a x^{n+1}}{n+1}\right )}+a x^n \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
n = symbols("n")
y = Function("y")
ode = Eq(a**2*x**(2*n) - a*n*x**(n - 1) - y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE a**2*x**(2*n) - a*n*x**(n - 1) - y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method