60.1.25 problem 25
Internal
problem
ID
[10039]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
25
Date
solved
:
Wednesday, March 05, 2025 at 08:05:14 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }+a y^{2}-b \,x^{2 \nu }-c \,x^{\nu -1}&=0 \end{align*}
✓ Maple. Time used: 0.001 (sec). Leaf size: 347
ode:=diff(y(x),x)+y(x)^2*a-b*x^(2*nu)-c*x^(nu-1) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (\left (\frac {\nu }{2}+1\right ) \sqrt {b}-\frac {\sqrt {a}\, c}{2}\right ) \operatorname {WhittakerM}\left (-\frac {\left (-2 \nu -2\right ) \sqrt {b}+\sqrt {a}\, c}{\sqrt {b}\, \left (2 \nu +2\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right )-c_{1} \sqrt {b}\, \left (\nu +1\right ) \operatorname {WhittakerW}\left (-\frac {\left (-2 \nu -2\right ) \sqrt {b}+\sqrt {a}\, c}{\sqrt {b}\, \left (2 \nu +2\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right )+\left (x^{\nu +1} b \sqrt {a}+\frac {\sqrt {a}\, c}{2}-\frac {\sqrt {b}\, \nu }{2}\right ) \left (\operatorname {WhittakerW}\left (-\frac {\sqrt {a}\, c}{\sqrt {b}\, \left (2 \nu +2\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {\sqrt {a}\, c}{\sqrt {b}\, \left (2 \nu +2\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right )\right )}{\sqrt {b}\, \left (\operatorname {WhittakerW}\left (-\frac {\sqrt {a}\, c}{\sqrt {b}\, \left (2 \nu +2\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {\sqrt {a}\, c}{\sqrt {b}\, \left (2 \nu +2\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right )\right ) a x}
\]
✓ Mathematica. Time used: 1.053 (sec). Leaf size: 928
ode=D[y[x],x] + a*y[x]^2 - b*x^(2*nu) - c*x^(nu-1)==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} \text {Solution too large to show}\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
nu = symbols("nu")
y = Function("y")
ode = Eq(a*y(x)**2 - b*x**(2*nu) - c*x**(nu - 1) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE a*y(x)**2 - b*x**(2*nu) - c*x**(nu - 1) + Derivative(y(x), x) cannot be solved by the lie group method