55.2.52 problem 52
Internal
problem
ID
[13278]
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
:
52
Date
solved
:
Wednesday, October 01, 2025 at 05:07:16 AM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} x^{2} y^{\prime }&=a \,x^{2} y^{2}+b x y+c \,x^{2 n}+s \,x^{n} \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 356
ode:=x^2*diff(y(x),x) = a*x^2*y(x)^2+b*x*y(x)+c*x^(2*n)+s*x^n;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (-\left (n +b +1\right ) \sqrt {c}+i \sqrt {a}\, s \right ) \operatorname {KummerM}\left (\frac {\left (1+b -n \right ) \sqrt {c}+i \sqrt {a}\, s}{2 \sqrt {c}\, n}, \frac {n +b +1}{n}, \frac {2 i \sqrt {a}\, \sqrt {c}\, x^{n}}{n}\right )+2 \operatorname {KummerU}\left (\frac {\left (1+b -n \right ) \sqrt {c}+i \sqrt {a}\, s}{2 \sqrt {c}\, n}, \frac {n +b +1}{n}, \frac {2 i \sqrt {a}\, \sqrt {c}\, x^{n}}{n}\right ) \sqrt {c}\, c_1 n -2 \left (\operatorname {KummerU}\left (\frac {\left (n +b +1\right ) \sqrt {c}+i \sqrt {a}\, s}{2 \sqrt {c}\, n}, \frac {n +b +1}{n}, \frac {2 i \sqrt {a}\, \sqrt {c}\, x^{n}}{n}\right ) c_1 +\operatorname {KummerM}\left (\frac {\left (n +b +1\right ) \sqrt {c}+i \sqrt {a}\, s}{2 \sqrt {c}\, n}, \frac {n +b +1}{n}, \frac {2 i \sqrt {a}\, \sqrt {c}\, x^{n}}{n}\right )\right ) \left (\frac {\left (1+b -n \right ) \sqrt {c}}{2}+i \sqrt {a}\, \left (x^{n} c +\frac {s}{2}\right )\right )}{2 \sqrt {c}\, x a \left (\operatorname {KummerU}\left (\frac {\left (n +b +1\right ) \sqrt {c}+i \sqrt {a}\, s}{2 \sqrt {c}\, n}, \frac {n +b +1}{n}, \frac {2 i \sqrt {a}\, \sqrt {c}\, x^{n}}{n}\right ) c_1 +\operatorname {KummerM}\left (\frac {\left (n +b +1\right ) \sqrt {c}+i \sqrt {a}\, s}{2 \sqrt {c}\, n}, \frac {n +b +1}{n}, \frac {2 i \sqrt {a}\, \sqrt {c}\, x^{n}}{n}\right )\right )}
\]
✓ Mathematica. Time used: 0.634 (sec). Leaf size: 819
ode=x^2*D[y[x],x]==a*x^2*y[x]^2+b*x*y[x]+c*x^(2*n)+s*x^n;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {i \sqrt {a} c_1 x^n \left (\sqrt {c} (b+n+1)-i \sqrt {a} s\right ) \operatorname {HypergeometricU}\left (\frac {b+3 n-\frac {i \sqrt {a} s}{\sqrt {c}}+1}{2 n},\frac {b+2 n+1}{n},-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )+c_1 n \left (i \sqrt {a} \sqrt {c} x^n+b+1\right ) \operatorname {HypergeometricU}\left (\frac {b+n-\frac {i \sqrt {a} s}{\sqrt {c}}+1}{2 n},\frac {b+n+1}{n},-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )+n \left (2 i \sqrt {a} \sqrt {c} x^n L_{-\frac {b+3 n-\frac {i \sqrt {a} s}{\sqrt {c}}+1}{2 n}}^{\frac {b+n+1}{n}}\left (-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )+\left (i \sqrt {a} \sqrt {c} x^n+b+1\right ) L_{-\frac {b+n-\frac {i \sqrt {a} s}{\sqrt {c}}+1}{2 n}}^{\frac {b+1}{n}}\left (-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )\right )}{a n x \left (c_1 \operatorname {HypergeometricU}\left (\frac {b+n-\frac {i \sqrt {a} s}{\sqrt {c}}+1}{2 n},\frac {b+n+1}{n},-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )+L_{-\frac {b+n-\frac {i \sqrt {a} s}{\sqrt {c}}+1}{2 n}}^{\frac {b+1}{n}}\left (-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )\right )}\\ y(x)&\to -\frac {\frac {\sqrt {a} x^n \left (\sqrt {a} s+i \sqrt {c} (b+n+1)\right ) \operatorname {HypergeometricU}\left (\frac {b+3 n-\frac {i \sqrt {a} s}{\sqrt {c}}+1}{2 n},\frac {b+2 n+1}{n},-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )}{n \operatorname {HypergeometricU}\left (\frac {b+n-\frac {i \sqrt {a} s}{\sqrt {c}}+1}{2 n},\frac {b+n+1}{n},-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )}+i \sqrt {a} \sqrt {c} x^n+b+1}{a x}\\ y(x)&\to -\frac {\frac {\sqrt {a} x^n \left (\sqrt {a} s+i \sqrt {c} (b+n+1)\right ) \operatorname {HypergeometricU}\left (\frac {b+3 n-\frac {i \sqrt {a} s}{\sqrt {c}}+1}{2 n},\frac {b+2 n+1}{n},-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )}{n \operatorname {HypergeometricU}\left (\frac {b+n-\frac {i \sqrt {a} s}{\sqrt {c}}+1}{2 n},\frac {b+n+1}{n},-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )}+i \sqrt {a} \sqrt {c} x^n+b+1}{a x} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
n = symbols("n")
s = symbols("s")
y = Function("y")
ode = Eq(-a*x**2*y(x)**2 - b*x*y(x) - c*x**(2*n) - s*x**n + x**2*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (a*x**2*y(x)**2 + b*x*y(x) + x**n*(c*x**n + s))/x**2 cannot be solved by the factorable group method