29.35.8 problem 1040
Internal
problem
ID
[5606]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
35
Problem
number
:
1040
Date
solved
:
Sunday, March 30, 2025 at 09:17:35 AM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{3}+{y^{\prime }}^{2}-y&=0 \end{align*}
✓ Maple. Time used: 0.039 (sec). Leaf size: 425
ode:=diff(y(x),x)^3+diff(y(x),x)^2-y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
-3 \sqrt {3}\, 2^{{1}/{3}} \int _{}^{y}\frac {\left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{{1}/{3}}}{3^{{2}/{3}} 2^{{2}/{3}}-\sqrt {3}\, 2^{{1}/{3}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{{1}/{3}}+3^{{1}/{3}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{{2}/{3}}}d \textit {\_a} +x -c_1 &= 0 \\
\frac {12 \sqrt {3}\, 2^{{1}/{3}} \int _{}^{y}-\frac {\left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{{1}/{3}}}{\left (i 2^{{1}/{3}} 3^{{5}/{6}}+3^{{1}/{3}} 2^{{1}/{3}}-2 \,3^{{1}/{6}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{{1}/{3}}\right ) \left (3^{{1}/{3}} 2^{{1}/{3}}+3^{{1}/{6}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{{1}/{3}}\right )}d \textit {\_a} +\left (1+i \sqrt {3}\right ) \left (x -c_1 \right )}{1+i \sqrt {3}} &= 0 \\
\frac {12 i \sqrt {3}\, 2^{{1}/{3}} \int _{}^{y}\frac {\left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{{1}/{3}}}{\left (3^{{1}/{3}} 2^{{1}/{3}}+3^{{1}/{6}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{{1}/{3}}\right ) \left (i 2^{{1}/{3}} 3^{{5}/{6}}-3^{{1}/{3}} 2^{{1}/{3}}+2 \,3^{{1}/{6}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{{1}/{3}}\right )}d \textit {\_a} +\left (x -c_1 \right ) \left (\sqrt {3}+i\right )}{\sqrt {3}+i} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 150.507 (sec). Leaf size: 510
ode=(D[y[x],x])^3 + (D[y[x],x])^2 -y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{-27 K[1]+3 \sqrt {3} \sqrt {K[1] (27 K[1]-4)}+2}}{2^{2/3} \left (-27 K[1]+3 \sqrt {3} \sqrt {K[1] (27 K[1]-4)}+2\right )^{2/3}+2 \sqrt [3]{-27 K[1]+3 \sqrt {3} \sqrt {K[1] (27 K[1]-4)}+2}+2 \sqrt [3]{2}}dK[1]\&\right ]\left [-\frac {x}{6}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{-27 K[2]+3 \sqrt {3} \sqrt {K[2] (27 K[2]-4)}+2}}{-i 2^{2/3} \sqrt {3} \left (-27 K[2]+3 \sqrt {3} \sqrt {K[2] (27 K[2]-4)}+2\right )^{2/3}+2^{2/3} \left (-27 K[2]+3 \sqrt {3} \sqrt {K[2] (27 K[2]-4)}+2\right )^{2/3}-4 \sqrt [3]{-27 K[2]+3 \sqrt {3} \sqrt {K[2] (27 K[2]-4)}+2}+2 i \sqrt [3]{2} \sqrt {3}+2 \sqrt [3]{2}}dK[2]\&\right ]\left [\frac {x}{12}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{-27 K[3]+3 \sqrt {3} \sqrt {K[3] (27 K[3]-4)}+2}}{i 2^{2/3} \sqrt {3} \left (-27 K[3]+3 \sqrt {3} \sqrt {K[3] (27 K[3]-4)}+2\right )^{2/3}+2^{2/3} \left (-27 K[3]+3 \sqrt {3} \sqrt {K[3] (27 K[3]-4)}+2\right )^{2/3}-4 \sqrt [3]{-27 K[3]+3 \sqrt {3} \sqrt {K[3] (27 K[3]-4)}+2}-2 i \sqrt [3]{2} \sqrt {3}+2 \sqrt [3]{2}}dK[3]\&\right ]\left [\frac {x}{12}+c_1\right ] \\
\end{align*}
✓ Sympy. Time used: 77.656 (sec). Leaf size: 454
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-y(x) + Derivative(y(x), x)**3 + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ - 6 \left (\sqrt {3} - i\right ) \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 27 y + 3 \sqrt {3} \sqrt {y \left (27 y - 4\right )} + 2}}{2^{\frac {2}{3}} \sqrt {3} \left (- 27 y + 3 \sqrt {3} \sqrt {y \left (27 y - 4\right )} + 2\right )^{\frac {2}{3}} + 2^{\frac {2}{3}} i \left (- 27 y + 3 \sqrt {3} \sqrt {y \left (27 y - 4\right )} + 2\right )^{\frac {2}{3}} - 2 \sqrt {3} \sqrt [3]{- 27 y + 3 \sqrt {3} \sqrt {y \left (27 y - 4\right )} + 2} + 2 i \sqrt [3]{- 27 y + 3 \sqrt {3} \sqrt {y \left (27 y - 4\right )} + 2} - 4 \sqrt [3]{2} i}\, dy = C_{1} - x, \ - 6 \left (\sqrt {3} + i\right ) \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 27 y + 3 \sqrt {3} \sqrt {y \left (27 y - 4\right )} + 2}}{2^{\frac {2}{3}} \sqrt {3} \left (- 27 y + 3 \sqrt {3} \sqrt {y \left (27 y - 4\right )} + 2\right )^{\frac {2}{3}} - 2^{\frac {2}{3}} i \left (- 27 y + 3 \sqrt {3} \sqrt {y \left (27 y - 4\right )} + 2\right )^{\frac {2}{3}} - 2 \sqrt {3} \sqrt [3]{- 27 y + 3 \sqrt {3} \sqrt {y \left (27 y - 4\right )} + 2} - 2 i \sqrt [3]{- 27 y + 3 \sqrt {3} \sqrt {y \left (27 y - 4\right )} + 2} + 4 \sqrt [3]{2} i}\, dy = C_{1} - x, \ \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 27 y + 3 \sqrt {3} \sqrt {y \left (27 y - 4\right )} + 2}}{2^{\frac {2}{3}} \left (- 27 y + 3 \sqrt {3} \sqrt {y \left (27 y - 4\right )} + 2\right )^{\frac {2}{3}} + 2 \sqrt [3]{- 27 y + 3 \sqrt {3} \sqrt {y \left (27 y - 4\right )} + 2} + 2 \sqrt [3]{2}}\, dy = C_{1} - \frac {x}{6}\right ]
\]