77.1.52 problem 71 (page 112)
Internal
problem
ID
[17863]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
71
(page
112)
Date
solved
:
Thursday, March 13, 2025 at 11:01:42 AM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{3}+y^{3}-3 y^{\prime } y&=0 \end{align*}
✓ Maple. Time used: 0.283 (sec). Leaf size: 248
ode:=diff(y(x),x)^3+y(x)^3-3*y(x)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
x -2 \left (\int _{}^{y}\frac {\left (-4 \textit {\_a}^{3}+4 \sqrt {\textit {\_a}^{3} \left (\textit {\_a}^{3}-4\right )}\right )^{{1}/{3}}}{\left (-4 \textit {\_a}^{3}+4 \sqrt {\textit {\_a}^{3} \left (\textit {\_a}^{3}-4\right )}\right )^{{2}/{3}}+4 \textit {\_a}}d \textit {\_a} \right )-c_{1} &= 0 \\
\frac {-4 \left (\int _{}^{y}\frac {\left (-4 \textit {\_a}^{3}+4 \sqrt {\textit {\_a}^{3} \left (\textit {\_a}^{3}-4\right )}\right )^{{1}/{3}}}{2 i \textit {\_a} \sqrt {3}-\left (-4 \textit {\_a}^{3}+4 \sqrt {\textit {\_a}^{3} \left (\textit {\_a}^{3}-4\right )}\right )^{{2}/{3}}+2 \textit {\_a}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}+x -c_{1}}{1+i \sqrt {3}} &= 0 \\
\frac {-4 \left (\int _{}^{y}\frac {\left (-4 \textit {\_a}^{3}+4 \sqrt {\textit {\_a}^{3} \left (\textit {\_a}^{3}-4\right )}\right )^{{1}/{3}}}{2 i \textit {\_a} \sqrt {3}+\left (-4 \textit {\_a}^{3}+4 \sqrt {\textit {\_a}^{3} \left (\textit {\_a}^{3}-4\right )}\right )^{{2}/{3}}-2 \textit {\_a}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}-x +c_{1}}{i \sqrt {3}-1} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 95.245 (sec). Leaf size: 383
ode=D[y[x],x]^3+y[x]^3-3*D[y[x],x]*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{\sqrt {K[1]^3 \left (K[1]^3-4\right )}-K[1]^3}}{2 K[1]+\sqrt [3]{2} \left (\sqrt {K[1]^3 \left (K[1]^3-4\right )}-K[1]^3\right )^{2/3}}dK[1]\&\right ]\left [\frac {x}{2^{2/3}}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{\sqrt {K[2]^3 \left (K[2]^3-4\right )}-K[2]^3}}{-2 \sqrt {3} K[2]-2 i K[2]+\sqrt [3]{2} \sqrt {3} \left (\sqrt {K[2]^3 \left (K[2]^3-4\right )}-K[2]^3\right )^{2/3}-i \sqrt [3]{2} \left (\sqrt {K[2]^3 \left (K[2]^3-4\right )}-K[2]^3\right )^{2/3}}dK[2]\&\right ]\left [c_1-\frac {i x}{2\ 2^{2/3}}\right ] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{\sqrt {K[3]^3 \left (K[3]^3-4\right )}-K[3]^3}}{-2 \sqrt {3} K[3]+2 i K[3]+\sqrt [3]{2} \sqrt {3} \left (\sqrt {K[3]^3 \left (K[3]^3-4\right )}-K[3]^3\right )^{2/3}+i \sqrt [3]{2} \left (\sqrt {K[3]^3 \left (K[3]^3-4\right )}-K[3]^3\right )^{2/3}}dK[3]\&\right ]\left [\frac {i x}{2\ 2^{2/3}}+c_1\right ] \\
\end{align*}
✓ Sympy. Time used: 3.468 (sec). Leaf size: 201
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(y(x)**3 - 3*y(x)*Derivative(y(x), x) + Derivative(y(x), x)**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} e^{\frac {\sqrt [3]{2} x \left (- \frac {2 i}{\sqrt [3]{1 + \sqrt {3} i}} + \frac {\sqrt [3]{2} \sqrt {3} \sqrt [3]{1 + \sqrt {3} i}}{2} + \frac {\sqrt [3]{2} i \sqrt [3]{1 + \sqrt {3} i}}{2}\right )}{\sqrt {3} - i}}, \ y{\left (x \right )} = C_{1} e^{\frac {\sqrt [3]{2} x \left (- \frac {\sqrt [3]{2} i \sqrt [3]{1 + \sqrt {3} i}}{2} + \frac {\sqrt [3]{2} \sqrt {3} \sqrt [3]{1 + \sqrt {3} i}}{2} + \frac {2 i}{\sqrt [3]{1 + \sqrt {3} i}}\right )}{\sqrt {3} + i}}, \ y{\left (x \right )} = C_{1} e^{- \sqrt [3]{2} x \left (\frac {1}{\sqrt [3]{1 + \sqrt {3} i}} + \frac {\sqrt [3]{2} \sqrt [3]{1 + \sqrt {3} i}}{2}\right )}\right ]
\]