60.1.521 problem 524
Internal
problem
ID
[10535]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
524
Date
solved
:
Monday, January 27, 2025 at 08:52:11 PM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{3}-2 y^{\prime } y+y^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.051 (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 &= 0 \\
-2^{{2}/{3}} \sqrt {3}\, \left (\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} \right )+x -c_{1} &= 0 \\
\frac {2 i 2^{{2}/{3}} \sqrt {3}\, \left (\int _{}^{y}\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 (x -c_{1} \right ) \left (-i+\sqrt {3}\right )}{-i+\sqrt {3}} &= 0 \\
\frac {2 i 2^{{2}/{3}} \sqrt {3}\, \left (\int _{}^{y}\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 (x -c_{1} \right ) \left (\sqrt {3}+i\right )}{\sqrt {3}+i} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.567 (sec). Leaf size: 427
DSolve[y[x]^2 - 2*y[x]*D[y[x],x] + D[y[x],x]^3==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*}