61.2.53 problem 53
Internal
problem
ID
[11980]
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
:
53
Date
solved
:
Sunday, March 30, 2025 at 09:31:51 PM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} x^{2} y^{\prime }&=c \,x^{2} y^{2}+\left (a \,x^{n}+b \right ) x y+\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 560
ode:=x^2*diff(y(x),x) = c*x^2*y(x)^2+(a*x^n+b)*x*y(x)+alpha*x^(2*n)+beta*x^n+gamma;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {\left (\sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 \alpha c}+\sqrt {a^{2}-4 \alpha c}\, n +\left (-b +n -1\right ) a +2 c \beta \right ) \operatorname {WhittakerM}\left (-\frac {-2 \sqrt {a^{2}-4 \alpha c}\, n +\left (b -n +1\right ) a -2 c \beta }{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )-2 \operatorname {WhittakerW}\left (-\frac {-2 \sqrt {a^{2}-4 \alpha c}\, n +\left (b -n +1\right ) a -2 c \beta }{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right ) c_1 n \sqrt {a^{2}-4 \alpha c}+\left (\operatorname {WhittakerW}\left (-\frac {\left (b -n +1\right ) a -2 c \beta }{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right ) c_1 +\operatorname {WhittakerM}\left (-\frac {\left (b -n +1\right ) a -2 c \beta }{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )\right ) \left (\left (a \,x^{n}+b -n +1\right ) \sqrt {a^{2}-4 \alpha c}+\left (a^{2}-4 \alpha c \right ) x^{n}+\left (b -n +1\right ) a -2 c \beta \right )}{2 \sqrt {a^{2}-4 \alpha c}\, \left (\operatorname {WhittakerW}\left (-\frac {\left (b -n +1\right ) a -2 c \beta }{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right ) c_1 +\operatorname {WhittakerM}\left (-\frac {\left (b -n +1\right ) a -2 c \beta }{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )\right ) c x}
\]
✓ Mathematica. Time used: 2.182 (sec). Leaf size: 2380
ode=x^2*D[y[x],x]==c*x^2*y[x]^2+(a*x^n+b)*x*y[x]+\[Alpha]*x^(2*n)+\[Beta]*x^n+\[Gamma];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Too large to display
✗ Sympy
from sympy import *
x = symbols("x")
Alpha = symbols("Alpha")
BETA = symbols("BETA")
Gamma = symbols("Gamma")
a = symbols("a")
b = symbols("b")
c = symbols("c")
n = symbols("n")
y = Function("y")
ode = Eq(-Alpha*x**(2*n) - BETA*x**n - Gamma - c*x**2*y(x)**2 + x**2*Derivative(y(x), x) - x*(a*x**n + b)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (Alpha*x**(2*n) + BETA*x**n + Gamma + c*x**2*y(x)**2 + x*(a*x**n + b)*y(x))/x**2 cannot be solved by the factorable group method