29.3.3 problem 57
Internal
problem
ID
[4665]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
3
Problem
number
:
57
Date
solved
:
Tuesday, March 04, 2025 at 07:01:43 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=a \,x^{n -1}+b \,x^{2 n}+c y^{2} \end{align*}
✓ Maple. Time used: 0.004 (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);
\[
y \left (x \right ) = -\frac {\left (\left (2+n \right ) \sqrt {b}-i \sqrt {c}\, a \right ) \operatorname {WhittakerM}\left (-\frac {\left (-2 n -2\right ) \sqrt {b}+i \sqrt {c}\, a}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {b}\, \sqrt {c}\, x^{n} x}{n +1}\right )-2 \sqrt {b}\, c_{1} \left (n +1\right ) \operatorname {WhittakerW}\left (-\frac {\left (-2 n -2\right ) \sqrt {b}+i \sqrt {c}\, a}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {b}\, \sqrt {c}\, x^{n} x}{n +1}\right )+\left (-\sqrt {b}\, n +i \sqrt {c}\, \left (2 x^{n} b x +a \right )\right ) \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {c}\, a}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {b}\, \sqrt {c}\, x^{n} x}{n +1}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {i \sqrt {c}\, a}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {b}\, \sqrt {c}\, x^{n} x}{n +1}\right )\right )}{2 \sqrt {b}\, \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {c}\, a}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {b}\, \sqrt {c}\, x^{n} x}{n +1}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {i \sqrt {c}\, a}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {b}\, \sqrt {c}\, x^{n} x}{n +1}\right )\right ) c x}
\]
✓ Mathematica. Time used: 1.153 (sec). Leaf size: 982
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*} \text {Solution too large to show}\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