29.23.18 problem 649
Internal
problem
ID
[5240]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
23
Problem
number
:
649
Date
solved
:
Monday, January 27, 2025 at 10:42:45 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} x \left (x^{2}-2 y^{2}\right ) y^{\prime }&=\left (2 x^{2}-y^{2}\right ) y \end{align*}
✓ Solution by Maple
Time used: 0.811 (sec). Leaf size: 1755
dsolve(x*(x^2-2*y(x)^2)*diff(y(x),x) = (2*x^2-y(x)^2)*y(x),y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= -\frac {6 x^{2} c_{1} \sqrt {3}\, \sqrt {2}\, \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}}{\left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{2}/{3}}+2 \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}+4\right ) \sqrt {\frac {\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{2}/{3}}+2 \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}+4}{\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}}}} \\
y \left (x \right ) &= \frac {6 x^{2} c_{1} \sqrt {3}\, \sqrt {2}\, \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}}{\left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{2}/{3}}+2 \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}+4\right ) \sqrt {\frac {\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{2}/{3}}+2 \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}+4}{\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}}}} \\
y \left (x \right ) &= \frac {24 x^{2} c_{1} \sqrt {3}\, \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}}{\left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}-2\right ) \sqrt {\frac {\left (-\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}+2\right ) \left (-2+i \left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}+2\right ) \sqrt {3}+\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}\right )}{\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}}}\, \left (-2+i \left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}+2\right ) \sqrt {3}+\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}\right )} \\
y \left (x \right ) &= \frac {24 x^{2} c_{1} \sqrt {3}\, \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}}{\left (-\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}+2\right ) \sqrt {\frac {\left (-\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}+2\right ) \left (-2+i \left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}+2\right ) \sqrt {3}+\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}\right )}{\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}}}\, \left (-2+i \left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}+2\right ) \sqrt {3}+\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}\right )} \\
y \left (x \right ) &= -\frac {24 x^{2} c_{1} \sqrt {3}\, \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}}{\sqrt {\frac {\left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}-2\right ) \left (2+i \left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}+2\right ) \sqrt {3}-\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}\right )}{\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}}}\, \left (\left (i \sqrt {3}-1\right ) \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}+2 i \sqrt {3}+2\right ) \left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}-2\right )} \\
y \left (x \right ) &= \frac {24 x^{2} c_{1} \sqrt {3}\, \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}}{\sqrt {\frac {\left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}-2\right ) \left (2+i \left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}+2\right ) \sqrt {3}-\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}\right )}{\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}}}\, \left (\left (i \sqrt {3}-1\right ) \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}+2 i \sqrt {3}+2\right ) \left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{{1}/{3}}-2\right )} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.288 (sec). Leaf size: 873
DSolve[x(x^2-2 y[x]^2)D[y[x],x]==(2 x^2-y[x]^2)y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\sqrt {-x^2+\frac {\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}{\sqrt [3]{2} 3^{2/3}}+\frac {\sqrt [3]{\frac {2}{3}} e^{2 c_1} x^2}{\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}} \\
y(x)\to \sqrt {-x^2+\frac {\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}{\sqrt [3]{2} 3^{2/3}}+\frac {\sqrt [3]{\frac {2}{3}} e^{2 c_1} x^2}{\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}} \\
y(x)\to -\frac {1}{2} \sqrt {-4 x^2+\left (\frac {2}{3}\right )^{2/3} \left (-1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}-\frac {2 \sqrt [3]{2} \left (\sqrt {3}-3 i\right ) e^{2 c_1} x^2}{3^{5/6} \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}} \\
y(x)\to \frac {1}{2} \sqrt {-4 x^2+\left (\frac {2}{3}\right )^{2/3} \left (-1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}-\frac {2 \sqrt [3]{2} \left (\sqrt {3}-3 i\right ) e^{2 c_1} x^2}{3^{5/6} \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}} \\
y(x)\to -\frac {1}{2} \sqrt {-4 x^2+i \left (\frac {2}{3}\right )^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}-\frac {2 \sqrt [3]{2} \left (\sqrt {3}+3 i\right ) e^{2 c_1} x^2}{3^{5/6} \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}} \\
y(x)\to \frac {1}{2} \sqrt {-4 x^2+i \left (\frac {2}{3}\right )^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}-\frac {2 \sqrt [3]{2} \left (\sqrt {3}+3 i\right ) e^{2 c_1} x^2}{3^{5/6} \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}} \\
\end{align*}