23.2.290 problem 309
Internal
problem
ID
[5645]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
2.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
SECOND
OR
HIGHER
DEGREE,
page
278
Problem
number
:
309
Date
solved
:
Tuesday, September 30, 2025 at 01:24:03 PM
CAS
classification
:
[_quadrature]
\begin{align*} 2 {y^{\prime }}^{3}+{y^{\prime }}^{2}-y&=0 \end{align*}
✓ Maple. Time used: 0.041 (sec). Leaf size: 388
ode:=2*diff(y(x),x)^3+diff(y(x),x)^2-y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
-6 \sqrt {3}\, \int _{}^{y}\frac {\left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}}{3^{{2}/{3}}-\sqrt {3}\, \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}+3^{{1}/{3}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{2}/{3}}}d \textit {\_a} +x -c_1 &= 0 \\
\frac {72 \int _{}^{y}-\frac {\left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}}{\left (3^{{1}/{3}}+3^{{1}/{6}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}\right ) \left (i 3^{{5}/{6}}-2 \,3^{{1}/{6}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}+3^{{1}/{3}}\right )}d \textit {\_a} +\left (-c_1 +x \right ) \sqrt {3}+3 i x -3 i c_1}{\sqrt {3}+3 i} &= 0 \\
\frac {-72 \int _{}^{y}\frac {\left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}}{\left (i 3^{{5}/{6}}-3^{{1}/{3}}+2 \,3^{{1}/{6}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}\right ) \left (3^{{1}/{3}}+3^{{1}/{6}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}\right )}d \textit {\_a} +\left (c_1 -x \right ) \sqrt {3}+3 i x -3 i c_1}{-\sqrt {3}+3 i} &= 0 \\
\end{align*}
✗ Mathematica
ode=2 (D[y[x],x])^3 + (D[y[x],x])^2 - y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Timed out
✓ Sympy. Time used: 49.792 (sec). Leaf size: 343
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-y(x) + 2*Derivative(y(x), x)**3 + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ - 6 \left (\sqrt {3} - i\right ) \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1}}{\left (\sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} - 1\right ) \left (\sqrt {3} \sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} + i \sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} + 2 i\right )}\, dy = C_{1} - x, \ - 6 \left (\sqrt {3} + i\right ) \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1}}{\left (\sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} - 1\right ) \left (\sqrt {3} \sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} - i \sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} - 2 i\right )}\, dy = C_{1} - x, \ \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1}}{\left (- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1\right )^{\frac {2}{3}} + \sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} + 1}\, dy = C_{1} - \frac {x}{6}\right ]
\]