60.7.32 problem 1622 (6.32)
Internal
problem
ID
[11582]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1622
(6.32)
Date
solved
:
Thursday, March 13, 2025 at 08:53:58 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} y^{\prime \prime }+\left (y+3 a \right ) y^{\prime }-y^{3}+a y^{2}+2 a^{2} y&=0 \end{align*}
✓ Maple. Time used: 0.076 (sec). Leaf size: 410
ode:=diff(diff(y(x),x),x)+(y(x)+3*a)*diff(y(x),x)-y(x)^3+y(x)^2*a+2*a^2*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \operatorname {RootOf}\left (\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} \right ) a +c_{2} a +{\mathrm e}^{-a x}\right ) {\mathrm e}^{-a x} \\
y &= \operatorname {RootOf}\left (-\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} \right ) a +2 c_{2} a +2 \,{\mathrm e}^{-a x}\right ) {\mathrm e}^{-a x} \\
y &= \operatorname {RootOf}\left (\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} \right ) a +2 c_{2} a +2 \,{\mathrm e}^{-a x}\right ) {\mathrm e}^{-a x} \\
\end{align*}
✓ Mathematica. Time used: 82.479 (sec). Leaf size: 88
ode=2*a^2*y[x] + a*y[x]^2 - y[x]^3 + (3*a + y[x])*D[y[x],x] + D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {c_1 \wp ''(x c_1+c_2;0,1)}{\wp (x c_1+c_2;0,1)} & a=0 \\ -\frac {e^{-a x} c_1 \wp ''\left (\frac {e^{-a x} c_1}{a}+c_2;0,1\right )}{\wp \left (\frac {e^{-a x} c_1}{a}+c_2;0,1\right )} & \text {True} \\ \end {array} \\ \end {array}
\]
✗ 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 - Derivative(y(x), (x, 2)))/(3*a + y(x)) cannot be solved by the factorable group method