61.2.10 problem 10
Internal
problem
ID
[11937]
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
:
10
Date
solved
:
Sunday, March 30, 2025 at 09:22:34 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=a \,x^{n} y^{2}+b m \,x^{m -1}-a \,b^{2} x^{n +2 m} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 512
ode:=diff(y(x),x) = a*x^n*y(x)^2+b*m*x^(m-1)-a*b^2*x^(n+2*m);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (-3 \left (m +2 n +2\right ) c_1 \left (\frac {\left (m +2 n +2\right ) \left (m +n +1\right ) x^{-\frac {3 m}{2}}}{3}+a \,x^{-\frac {m}{2}} b x \,x^{n} \left (m +\frac {4 n}{3}+\frac {4}{3}\right )\right ) \operatorname {WhittakerM}\left (\frac {m +2 n +2}{2 n +2 m +2}, \frac {2 m +3 n +3}{2 n +2 m +2}, -\frac {2 a b \,x^{m} x^{n} x}{m +n +1}\right )+2 \left (-\frac {\left (m +2 n +2\right ) \left (m +n +1\right ) x^{-\frac {3 m}{2}}}{2}+a b x \left (\left (-\frac {m}{2}-n -1\right ) x^{-\frac {m}{2}}+a \,x^{\frac {m}{2}} b x \,x^{n}\right ) x^{n}\right ) c_1 \left (m +n +1\right ) \operatorname {WhittakerM}\left (-\frac {m}{2 n +2 m +2}, \frac {2 m +3 n +3}{2 n +2 m +2}, -\frac {2 a b \,x^{m} x^{n} x}{m +n +1}\right )+2 \left (\left (m +\frac {3 n}{2}+\frac {3}{2}\right ) x^{-\frac {3 m}{2}-n -1} {\mathrm e}^{\frac {a b \,x^{m +n +1}}{m +n +1}} \left (m +2 n +2\right )^{2} c_1 \left (-\frac {2 a b \,x^{m +n +1}}{m +n +1}\right )^{\frac {3 m +4 n +4}{2 n +2 m +2}}+a b \,x^{m +n +1} {\mathrm e}^{-\frac {a b \,x^{m +n +1}}{m +n +1}}\right ) x \,x^{n}\right ) x^{-n}}{2 a x \left (-\frac {c_1 \,x^{-\frac {3 m}{2}} \left (m +2 n +2\right )^{2} \operatorname {WhittakerM}\left (\frac {m +2 n +2}{2 n +2 m +2}, \frac {2 m +3 n +3}{2 n +2 m +2}, -\frac {2 a b \,x^{m +n +1}}{m +n +1}\right )}{2}+c_1 \left (a \,x^{1+n -\frac {m}{2}} b -\frac {x^{-\frac {3 m}{2}} \left (m +2 n +2\right )}{2}\right ) \left (m +n +1\right ) \operatorname {WhittakerM}\left (-\frac {m}{2 n +2 m +2}, \frac {2 m +3 n +3}{2 n +2 m +2}, -\frac {2 a b \,x^{m +n +1}}{m +n +1}\right )+x^{1+n} {\mathrm e}^{-\frac {a b \,x^{m +n +1}}{m +n +1}}\right )}
\]
✓ Mathematica. Time used: 1.39 (sec). Leaf size: 306
ode=D[y[x],x]==a*x^n*y[x]^2+b*m*x^(m-1)-a*b^2*x^(n+2*m);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {2^{\frac {n+1}{m+n+1}} (m+n+1) \left (-\frac {a b x^{m+n+1}}{m+n+1}\right )^{\frac {n+1}{m+n+1}} \left (a b x^m-c_1 e^{\frac {2 a b x^{m+n+1}}{m+n+1}}\right )-a b c_1 x^{m+n+1} \Gamma \left (\frac {n+1}{m+n+1},-\frac {2 a b x^{m+n+1}}{m+n+1}\right )}{a \left (2^{\frac {n+1}{m+n+1}} (m+n+1) \left (-\frac {a b x^{m+n+1}}{m+n+1}\right )^{\frac {n+1}{m+n+1}}-c_1 x^{n+1} \Gamma \left (\frac {n+1}{m+n+1},-\frac {2 a b x^{m+n+1}}{m+n+1}\right )\right )} \\
y(x)\to b x^m-\frac {b 2^{\frac {n+1}{m+n+1}} x^m e^{\frac {2 a b x^{m+n+1}}{m+n+1}} \left (-\frac {a b x^{m+n+1}}{m+n+1}\right )^{-\frac {m}{m+n+1}}}{\Gamma \left (\frac {n+1}{m+n+1},-\frac {2 a b x^{m+n+1}}{m+n+1}\right )} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
m = symbols("m")
n = symbols("n")
y = Function("y")
ode = Eq(a*b**2*x**(2*m + n) - a*x**n*y(x)**2 - b*m*x**(m - 1) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE a*b**2*x**(2*m + n) - a*x**n*y(x)**2 - b*m*x**(m - 1) + Derivative(y(x), x) cannot be solved by the lie group method