23.4.285 problem 288
Internal
problem
ID
[6587]
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
:
288
Date
solved
:
Tuesday, September 30, 2025 at 03:22:19 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} y^{3}+\left (y^{2}+{y^{\prime }}^{2}\right ) y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.146 (sec). Leaf size: 163
ode:=y(x)^3+(y(x)^2+diff(y(x),x)^2)*diff(diff(y(x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
y &= \frac {\sqrt {c_1 +\tan \left (\sqrt {3}\, x \right )}\, {\mathrm e}^{\frac {\sqrt {3}\, \int \frac {\sqrt {\left (9 c_1^{2}+12\right ) \sec \left (\sqrt {3}\, x \right )^{2}+3 c_1^{2}+6 c_1 \tan \left (\sqrt {3}\, x \right )-3}}{c_1 +\tan \left (\sqrt {3}\, x \right )}d x}{6}+c_2}}{\left (\sec \left (\sqrt {3}\, x \right )^{2}\right )^{{1}/{4}}} \\
y &= \frac {\sqrt {c_1 +\tan \left (\sqrt {3}\, x \right )}\, {\mathrm e}^{-\frac {\sqrt {3}\, \int \frac {\sqrt {\left (9 c_1^{2}+12\right ) \sec \left (\sqrt {3}\, x \right )^{2}+3 c_1^{2}+6 c_1 \tan \left (\sqrt {3}\, x \right )-3}}{c_1 +\tan \left (\sqrt {3}\, x \right )}d x}{6}+c_2}}{\left (\sec \left (\sqrt {3}\, x \right )^{2}\right )^{{1}/{4}}} \\
\end{align*}
✓ Mathematica. Time used: 37.655 (sec). Leaf size: 369
ode=y[x]^3 + (y[x]^2 + D[y[x],x]^2)*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {c_2 \exp \left (-\frac {\arctan \left (\frac {1+2 \text {InverseFunction}\left [\frac {\left (\sqrt {3}-i\right ) \arctan \left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (\sqrt {3}+i\right ) \arctan \left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1+i \sqrt {3}\right )}}\&\right ][-x+c_1]{}^2}{\sqrt {3}}\right )}{2 \sqrt {3}}\right )}{\sqrt [4]{\text {InverseFunction}\left [\frac {\left (\sqrt {3}-i\right ) \arctan \left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (\sqrt {3}+i\right ) \arctan \left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1+i \sqrt {3}\right )}}\&\right ][-x+c_1]{}^4+\text {InverseFunction}\left [\frac {\left (\sqrt {3}-i\right ) \arctan \left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1-i \sqrt {3}\right )}}+\frac {\left (\sqrt {3}+i\right ) \arctan \left (\frac {\text {$\#$1}}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}{\sqrt {6 \left (1+i \sqrt {3}\right )}}\&\right ][-x+c_1]{}^2+1}} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((y(x)**2 + Derivative(y(x), x)**2)*Derivative(y(x), (x, 2)) + y(x)**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE sqrt(-(y(x) + Derivative(y(x), (x, 2)))/Derivative(y(x), (x, 2))