23.1.64 problem 58
Internal
problem
ID
[4671]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
1.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
FIRST
DEGREE,
page
223
Problem
number
:
58
Date
solved
:
Tuesday, September 30, 2025 at 07:55:17 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=a \,x^{n -1}+b \,x^{2 n}+c y^{2} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 340
ode:=diff(y(x),x) = a*x^(n-1)+b*x^(2*n)+c*y(x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {\left (\left (n +2\right ) \sqrt {b}-i \sqrt {c}\, a \right ) \operatorname {WhittakerM}\left (-\frac {\left (-2 n -2\right ) \sqrt {b}+i \sqrt {c}\, a}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {b}\, \sqrt {c}\, x^{n} x}{n +1}\right )-2 \sqrt {b}\, c_1 \left (n +1\right ) \operatorname {WhittakerW}\left (-\frac {\left (-2 n -2\right ) \sqrt {b}+i \sqrt {c}\, a}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {b}\, \sqrt {c}\, x^{n} x}{n +1}\right )+\left (-\sqrt {b}\, n +i \sqrt {c}\, \left (2 x^{n} b x +a \right )\right ) \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {c}\, a}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {b}\, \sqrt {c}\, x^{n} x}{n +1}\right ) c_1 +\operatorname {WhittakerM}\left (-\frac {i \sqrt {c}\, a}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {b}\, \sqrt {c}\, x^{n} x}{n +1}\right )\right )}{2 \sqrt {b}\, \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {c}\, a}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {b}\, \sqrt {c}\, x^{n} x}{n +1}\right ) c_1 +\operatorname {WhittakerM}\left (-\frac {i \sqrt {c}\, a}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {b}\, \sqrt {c}\, x^{n} x}{n +1}\right )\right ) c x}
\]
✓ Mathematica. Time used: 0.659 (sec). Leaf size: 764
ode=D[y[x],x]==a*x^(n-1)+b*x^(2*n)+c*y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {x^n \left (\sqrt {b} c_1 (n+1) \sqrt {-(n+1)^2} \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} a}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_1 \left (a \sqrt {c} (n+1)+\sqrt {b} \sqrt {-(n+1)^2} n\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} a}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {3 n+2}{n+1}\right ),\frac {n}{n+1}+1,\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+\sqrt {b} (n+1) \sqrt {-(n+1)^2} \left (L_{-\frac {\sqrt {c} a}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+2 L_{-\frac {\sqrt {c} a}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {3 n+2}{2 n+2}}^{\frac {n}{n+1}}\left (\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )\right )}{\sqrt {c} (n+1)^2 \left (L_{-\frac {\sqrt {c} a}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} a}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )}\\ y(x)&\to \frac {x^n \left (-\frac {\left (a \sqrt {c} (n+1)+\sqrt {b} \sqrt {-(n+1)^2} n\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} a}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}+2\right ),\frac {n}{n+1}+1,\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )}{\operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} a}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {b} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )}-\sqrt {b} \sqrt {-(n+1)^2} (n+1)\right )}{\sqrt {c} (n+1)^2} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
n = symbols("n")
y = Function("y")
ode = Eq(-a*x**(n - 1) - b*x**(2*n) - c*y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -a*x**(n - 1) - b*x**(2*n) - c*y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method