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
:
Monday, January 27, 2025 at 12:13:12 PM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{3}-2 y^{\prime } y+y^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 285
dsolve(diff(y(x),x)^3-2*y(x)*diff(y(x),x)+y(x)^2 = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= 0 \\
-2^{{2}/{3}} \sqrt {3}\, \left (\int _{}^{y \left (x \right )}\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} \right )+x -c_{1} &= 0 \\
\frac {2 i 2^{{2}/{3}} \sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {\left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{{1}/{3}}}{2 i \sqrt {3}\, \textit {\_a} -2^{{1}/{3}} \left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{{2}/{3}}+2 \textit {\_a}}d \textit {\_a} \right )+\left (-c_{1} +x \right ) \left (-i+\sqrt {3}\right )}{-i+\sqrt {3}} &= 0 \\
\frac {2 i 2^{{2}/{3}} \sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {\left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{{1}/{3}}}{2 i \sqrt {3}\, \textit {\_a} +2^{{1}/{3}} \left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{{2}/{3}}-2 \textit {\_a}}d \textit {\_a} \right )+\left (-c_{1} +x \right ) \left (\sqrt {3}+i\right )}{\sqrt {3}+i} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.592 (sec). Leaf size: 427
DSolve[(D[y[x],x])^3 -2*y[x]*D[y[x],x]+y[x]^2==0,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*}