60.7.30 problem 1620 (6.30)
Internal
problem
ID
[11580]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1620
(6.30)
Date
solved
:
Wednesday, March 05, 2025 at 02:32:46 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} y^{\prime \prime }+y y^{\prime }-y^{3}&=0 \end{align*}
✓ Maple. Time used: 0.059 (sec). Leaf size: 332
ode:=diff(diff(y(x),x),x)+y(x)*diff(y(x),x)-y(x)^3 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
2 \left (\int _{}^{y}\frac {\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{1}/{3}}}{\textit {\_a}^{4}-\textit {\_a}^{2} \left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{1}/{3}}+\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{2}/{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
-4 \left (\int _{}^{y}\frac {\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{1}/{3}}}{-i \sqrt {3}\, \textit {\_a}^{4}+i \left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{2}/{3}} \sqrt {3}+\textit {\_a}^{4}+2 \textit {\_a}^{2} \left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{1}/{3}}+\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{2}/{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
4 \left (\int _{}^{y}-\frac {\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{1}/{3}}}{i \sqrt {3}\, \textit {\_a}^{4}-i \left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{2}/{3}} \sqrt {3}+\textit {\_a}^{4}+2 \textit {\_a}^{2} \left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{1}/{3}}+\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{2}/{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 52.756 (sec). Leaf size: 1534
ode=-y[x]^3 + y[x]*D[y[x],x] + D[y[x],{x,2}] == 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")
y = Function("y")
ode = Eq(-y(x)**3 + y(x)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -(y(x)**3 - Derivative(y(x), (x, 2)))/y(x) + Derivative(y(x), x) cannot be solved by the factorable group method