23.4.22 problem 22
Internal
problem
ID
[6324]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
4.
THE
NONLINEAR
EQUATION
OF
SECOND
ORDER,
page
380
Problem
number
:
22
Date
solved
:
Friday, October 03, 2025 at 02:01:03 AM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} 2 a^{2} y+a y^{2}+\left (3 a +y\right ) y^{\prime }+y^{\prime \prime }&=y^{3} \end{align*}
✓ Maple. Time used: 0.030 (sec). Leaf size: 416
ode:=2*a^2*y(x)+a*y(x)^2+(3*a+y(x))*diff(y(x),x)+diff(diff(y(x),x),x) = y(x)^3;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \operatorname {RootOf}\left (\int _{}^{\textit {\_Z}}-\frac {-\textit {\_f}^{8}+c_1 \,\textit {\_f}^{2}-{\left (\left (-\textit {\_f}^{6}+c_1 \right )^{2} \left (\sqrt {\frac {c_1}{-\textit {\_f}^{6}+c_1}}-1\right )\right )}^{{2}/{3}}}{\left (-\textit {\_f}^{6}+c_1 \right ) {\left (\left (-\textit {\_f}^{6}+c_1 \right )^{2} \left (\sqrt {\frac {c_1}{-\textit {\_f}^{6}+c_1}}-1\right )\right )}^{{1}/{3}}}d \textit {\_f} a +c_2 a +{\mathrm e}^{-a x}\right ) {\mathrm e}^{-a x} \\
y &= \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}\frac {-i \sqrt {3}\, \textit {\_f}^{8}+\textit {\_f}^{8}+i \sqrt {3}\, c_1 \,\textit {\_f}^{2}+i \sqrt {3}\, {\left (\left (-\textit {\_f}^{6}+c_1 \right )^{2} \left (\sqrt {\frac {c_1}{-\textit {\_f}^{6}+c_1}}-1\right )\right )}^{{2}/{3}}-c_1 \,\textit {\_f}^{2}+{\left (\left (-\textit {\_f}^{6}+c_1 \right )^{2} \left (\sqrt {\frac {c_1}{-\textit {\_f}^{6}+c_1}}-1\right )\right )}^{{2}/{3}}}{\left (-\textit {\_f}^{6}+c_1 \right ) {\left (\left (-\textit {\_f}^{6}+c_1 \right )^{2} \left (\sqrt {\frac {c_1}{-\textit {\_f}^{6}+c_1}}-1\right )\right )}^{{1}/{3}}}d \textit {\_f} a +2 c_2 a +2 \,{\mathrm e}^{-a x}\right ) {\mathrm e}^{-a x} \\
y &= \operatorname {RootOf}\left (\int _{}^{\textit {\_Z}}\frac {-i \sqrt {3}\, \textit {\_f}^{8}-\textit {\_f}^{8}+i \sqrt {3}\, c_1 \,\textit {\_f}^{2}+i \sqrt {3}\, {\left (\left (-\textit {\_f}^{6}+c_1 \right )^{2} \left (\sqrt {\frac {c_1}{-\textit {\_f}^{6}+c_1}}-1\right )\right )}^{{2}/{3}}+c_1 \,\textit {\_f}^{2}-{\left (\left (-\textit {\_f}^{6}+c_1 \right )^{2} \left (\sqrt {\frac {c_1}{-\textit {\_f}^{6}+c_1}}-1\right )\right )}^{{2}/{3}}}{\left (-\textit {\_f}^{6}+c_1 \right ) {\left (\left (-\textit {\_f}^{6}+c_1 \right )^{2} \left (\sqrt {\frac {c_1}{-\textit {\_f}^{6}+c_1}}-1\right )\right )}^{{1}/{3}}}d \textit {\_f} a +2 c_2 a +2 \,{\mathrm e}^{-a x}\right ) {\mathrm e}^{-a x} \\
\end{align*}
✓ Mathematica. Time used: 81.724 (sec). Leaf size: 88
ode=2*a^2*y[x] + a*y[x]^2 + (3*a + y[x])*D[y[x],x] + D[y[x],{x,2}] == y[x]^3;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {c_1 \wp '(x c_1+c_2;0,1)}{\wp (x c_1+c_2;0,1)} & a=0 \\ -\frac {e^{-a x} c_1 \wp '\left (\frac {e^{-a x} c_1}{a}+c_2;0,1\right )}{\wp \left (\frac {e^{-a x} c_1}{a}+c_2;0,1\right )} & \text {True} \\ \end {array} \\ \end {array} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(2*a**2*y(x) + a*y(x)**2 + (3*a + y(x))*Derivative(y(x), x) - y(x)**3 + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (-2*a**2*y(x) - a*y(x)**2 + y(x)**3 - Deri