23.4.52 problem 52
Internal
problem
ID
[6354]
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
:
52
Date
solved
:
Tuesday, September 30, 2025 at 02:52:24 PM
CAS
classification
:
[[_2nd_order, _with_exponential_symmetries], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1]]
\begin{align*} \left ({\mathrm e}^{2 y}+x \right ) {y^{\prime }}^{3}+y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.022 (sec). Leaf size: 577
ode:=(exp(2*y(x))+x)*diff(y(x),x)^3+diff(diff(y(x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\ln \left (2\right )+\ln \left (\frac {\left (6 c_2 x -6 c_1 +c_2^{3}+2 \sqrt {-3 c_1 \,c_2^{3}-3 x^{2} c_2^{2}-18 c_2 x c_1 -16 x^{3}+9 c_1^{2}}\right )^{{2}/{3}}+c_2 \left (6 c_2 x -6 c_1 +c_2^{3}+2 \sqrt {-3 c_1 \,c_2^{3}-3 x^{2} c_2^{2}-18 c_2 x c_1 -16 x^{3}+9 c_1^{2}}\right )^{{1}/{3}}+c_2^{2}+4 x}{\left (6 c_2 x -6 c_1 +c_2^{3}+2 \sqrt {-3 c_1 \,c_2^{3}-3 x^{2} c_2^{2}-18 c_2 x c_1 -16 x^{3}+9 c_1^{2}}\right )^{{1}/{3}}}\right ) \\
y &= -2 \ln \left (2\right )+\ln \left (\frac {i \sqrt {3}\, c_2^{2}-i \sqrt {3}\, \left (6 c_2 x -6 c_1 +c_2^{3}+2 \sqrt {-3 c_1 \,c_2^{3}-3 x^{2} c_2^{2}-18 c_2 x c_1 -16 x^{3}+9 c_1^{2}}\right )^{{2}/{3}}+4 i \sqrt {3}\, x -c_2^{2}+2 c_2 \left (6 c_2 x -6 c_1 +c_2^{3}+2 \sqrt {-3 c_1 \,c_2^{3}-3 x^{2} c_2^{2}-18 c_2 x c_1 -16 x^{3}+9 c_1^{2}}\right )^{{1}/{3}}-\left (6 c_2 x -6 c_1 +c_2^{3}+2 \sqrt {-3 c_1 \,c_2^{3}-3 x^{2} c_2^{2}-18 c_2 x c_1 -16 x^{3}+9 c_1^{2}}\right )^{{2}/{3}}-4 x}{\left (6 c_2 x -6 c_1 +c_2^{3}+2 \sqrt {-3 c_1 \,c_2^{3}-3 x^{2} c_2^{2}-18 c_2 x c_1 -16 x^{3}+9 c_1^{2}}\right )^{{1}/{3}}}\right ) \\
y &= -2 \ln \left (2\right )+\ln \left (\frac {\left (i \sqrt {3}-1\right ) \left (6 c_2 x -6 c_1 +c_2^{3}+2 \sqrt {-3 c_1 \,c_2^{3}-3 x^{2} c_2^{2}-18 c_2 x c_1 -16 x^{3}+9 c_1^{2}}\right )^{{2}/{3}}+2 c_2 \left (6 c_2 x -6 c_1 +c_2^{3}+2 \sqrt {-3 c_1 \,c_2^{3}-3 x^{2} c_2^{2}-18 c_2 x c_1 -16 x^{3}+9 c_1^{2}}\right )^{{1}/{3}}+4 \left (-i \sqrt {3}-1\right ) \left (x +\frac {c_2^{2}}{4}\right )}{\left (6 c_2 x -6 c_1 +c_2^{3}+2 \sqrt {-3 c_1 \,c_2^{3}-3 x^{2} c_2^{2}-18 c_2 x c_1 -16 x^{3}+9 c_1^{2}}\right )^{{1}/{3}}}\right ) \\
\end{align*}
✗ Mathematica
ode=(E^(2*y[x]) + x)*D[y[x],x]^3 + D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Not solved
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((x + exp(2*y(x)))*Derivative(y(x), 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))/(x + exp(2*y(x))))**(1/3)/2 - sqrt(3)