29.35.3 problem 1034
Internal
problem
ID
[5601]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
35
Problem
number
:
1034
Date
solved
:
Sunday, March 30, 2025 at 09:05:43 AM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{3}-2 y y^{\prime }+y^{2}&=0 \end{align*}
✓ Maple. Time used: 0.040 (sec). Leaf size: 288
ode:=diff(y(x),x)^3-2*y(x)*diff(y(x),x)+y(x)^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
-2^{{2}/{3}} \sqrt {3}\, \int _{}^{y}\frac {\left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{{1}/{3}}}{2^{{1}/{3}} \left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{{2}/{3}}+4 \textit {\_a}}d \textit {\_a} +x -c_1 &= 0 \\
\frac {2 \,2^{{2}/{3}} \sqrt {3}\, \int _{}^{y}\frac {\left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{{1}/{3}}}{2^{{1}/{3}} \left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{{2}/{3}}-2 i \sqrt {3}\, \textit {\_a} -2 \textit {\_a}}d \textit {\_a} +\left (1+i \sqrt {3}\right ) \left (x -c_1 \right )}{1+i \sqrt {3}} &= 0 \\
\frac {2 i 2^{{2}/{3}} \sqrt {3}\, \int _{}^{y}\frac {\left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{{1}/{3}}}{2^{{1}/{3}} \left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{{2}/{3}}+2 i \sqrt {3}\, \textit {\_a} -2 \textit {\_a}}d \textit {\_a} +\left (x -c_1 \right ) \left (\sqrt {3}+i\right )}{\sqrt {3}+i} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.592 (sec). Leaf size: 427
ode=(D[y[x],x])^3 -2*y[x]*D[y[x],x]+y[x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\int \frac {\sqrt [3]{\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2}}{\sqrt [3]{2} \left (\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2\right )^{2/3}+4 \sqrt [3]{3} \text {$\#$1}}d\text {$\#$1}\&\right ]\left [\frac {x}{6^{2/3}}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\int \frac {\sqrt [3]{\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2}}{\sqrt [3]{2} 3^{2/3} \left (\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2\right )^{2/3}-\sqrt [3]{2} \sqrt [6]{3} i \left (\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2\right )^{2/3}-12 \text {$\#$1}-4 i \text {$\#$1} \sqrt {3}}d\text {$\#$1}\&\right ]\left [c_1-\frac {i x}{2\ 2^{2/3} 3^{5/6}}\right ] \\
y(x)\to \text {InverseFunction}\left [\int \frac {\sqrt [3]{\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2}}{\sqrt [3]{2} 3^{2/3} \left (\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2\right )^{2/3}+\sqrt [3]{2} \sqrt [6]{3} i \left (\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2\right )^{2/3}-12 \text {$\#$1}+4 i \text {$\#$1} \sqrt {3}}d\text {$\#$1}\&\right ]\left [\frac {i x}{2\ 2^{2/3} 3^{5/6}}+c_1\right ] \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 59.230 (sec). Leaf size: 320
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(y(x)**2 - 2*y(x)*Derivative(y(x), x) + Derivative(y(x), x)**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ - 3 \cdot 2^{\frac {2}{3}} \left (\sqrt {3} - i\right ) \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{9 y^{2} + \sqrt {3} \sqrt {y^{3} \left (27 y - 32\right )}}}{- 8 \cdot 3^{\frac {2}{3}} y i + \sqrt [3]{2} \cdot 3^{\frac {5}{6}} \left (9 y^{2} + \sqrt {3} \sqrt {y^{3} \left (27 y - 32\right )}\right )^{\frac {2}{3}} + \sqrt [3]{6} i \left (9 y^{2} + \sqrt {3} \sqrt {y^{3} \left (27 y - 32\right )}\right )^{\frac {2}{3}}}\, dy = C_{1} - x, \ - 3 \cdot 2^{\frac {2}{3}} \left (\sqrt {3} + i\right ) \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{9 y^{2} + \sqrt {3} \sqrt {y^{3} \left (27 y - 32\right )}}}{8 \cdot 3^{\frac {2}{3}} y i + \sqrt [3]{2} \cdot 3^{\frac {5}{6}} \left (9 y^{2} + \sqrt {3} \sqrt {y^{3} \left (27 y - 32\right )}\right )^{\frac {2}{3}} - \sqrt [3]{6} i \left (9 y^{2} + \sqrt {3} \sqrt {y^{3} \left (27 y - 32\right )}\right )^{\frac {2}{3}}}\, dy = C_{1} - x, \ 6 \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{9 y^{2} + \sqrt {3} \sqrt {y^{3} \left (27 y - 32\right )}}}{4 \sqrt [3]{18} y + \sqrt [3]{12} \left (9 y^{2} + \sqrt {3} \sqrt {y^{3} \left (27 y - 32\right )}\right )^{\frac {2}{3}}}\, dy = C_{1} - x\right ]
\]