29.24.21 problem 683
Internal
problem
ID
[5274]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
24
Problem
number
:
683
Date
solved
:
Monday, January 27, 2025 at 10:55:43 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} 2 y^{3} y^{\prime }&=x^{3}-x y^{2} \end{align*}
✓ Solution by Maple
Time used: 1.092 (sec). Leaf size: 761
dsolve(2*y(x)^3*diff(y(x),x) = x^3-x*y(x)^2,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {\frac {c_{1}^{2} x^{4}-c_{1} x^{2} \left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{1}/{3}}+\left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{2}/{3}}}{\left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{1}/{3}}}}}{2 \sqrt {c_{1}}} \\
y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {\frac {c_{1}^{2} x^{4}-c_{1} x^{2} \left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{1}/{3}}+\left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{2}/{3}}}{\left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{1}/{3}}}}}{2 \sqrt {c_{1}}} \\
y \left (x \right ) &= -\frac {\sqrt {\frac {i \sqrt {3}\, c_{1}^{2} x^{4}-c_{1}^{2} x^{4}-i \sqrt {3}\, \left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{2}/{3}}-2 c_{1} x^{2} \left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{1}/{3}}-\left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{2}/{3}}}{\left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{1}/{3}}}}}{2 \sqrt {c_{1}}} \\
y \left (x \right ) &= \frac {\sqrt {\frac {i \sqrt {3}\, c_{1}^{2} x^{4}-c_{1}^{2} x^{4}-i \sqrt {3}\, \left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{2}/{3}}-2 c_{1} x^{2} \left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{1}/{3}}-\left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{2}/{3}}}{\left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{1}/{3}}}}}{2 \sqrt {c_{1}}} \\
y \left (x \right ) &= -\frac {\sqrt {\frac {-i \sqrt {3}\, c_{1}^{2} x^{4}-c_{1}^{2} x^{4}+i \sqrt {3}\, \left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{2}/{3}}-2 c_{1} x^{2} \left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{1}/{3}}-\left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{2}/{3}}}{\left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{1}/{3}}}}}{2 \sqrt {c_{1}}} \\
y \left (x \right ) &= \frac {\sqrt {\frac {-i \sqrt {3}\, c_{1}^{2} x^{4}-c_{1}^{2} x^{4}+i \sqrt {3}\, \left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{2}/{3}}-2 c_{1} x^{2} \left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{1}/{3}}-\left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{2}/{3}}}{\left (2+x^{6} c_{1}^{3}+2 \sqrt {x^{6} c_{1}^{3}+1}\right )^{{1}/{3}}}}}{2 \sqrt {c_{1}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.137 (sec). Leaf size: 714
DSolve[2*y[x]^3*D[y[x],x]==x^3-x*y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\sqrt {\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-x^2+\frac {x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}}}{\sqrt {2}} \\
y(x)\to \frac {\sqrt {\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-x^2+\frac {x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}}}{\sqrt {2}} \\
y(x)\to -\frac {1}{2} \sqrt {\left (-1-i \sqrt {3}\right ) \sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-2 x^2+\frac {i \left (\sqrt {3}+i\right ) x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}} \\
y(x)\to \frac {1}{2} \sqrt {\left (-1-i \sqrt {3}\right ) \sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-2 x^2+\frac {i \left (\sqrt {3}+i\right ) x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}} \\
y(x)\to -\frac {1}{2} \sqrt {i \left (\sqrt {3}+i\right ) \sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-2 x^2+\frac {\left (-1-i \sqrt {3}\right ) x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}} \\
y(x)\to \frac {1}{2} \sqrt {i \left (\sqrt {3}+i\right ) \sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-2 x^2+\frac {\left (-1-i \sqrt {3}\right ) x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}} \\
\end{align*}