Internal problem ID [9515]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 9, system of higher order odes
Problem number: 1936.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=x \relax (t ) \left (y \relax (t )^{2}-z \relax (t )^{2}\right )\\ y^{\prime }\relax (t )&=-y \relax (t ) \left (z \relax (t )^{2}+x \relax (t )^{2}\right )\\ z^{\prime }\relax (t )&=z \relax (t ) \left (x \relax (t )^{2}+y \relax (t )^{2}\right ) \end {align*}
✓ Solution by Maple
Time used: 0.875 (sec). Leaf size: 712
dsolve({diff(x(t),t)=x(t)*(y(t)^2-z(t)^2),diff(y(t),t)=-y(t)*(z(t)^2+x(t)^2),diff(z(t),t)=z(t)*(x(t)^2+y(t)^2)},{x(t), y(t), z(t)}, singsol=all)
\begin{align*} \{x \relax (t ) = 0\} \\ \{y \relax (t ) = 0\} \\ \{z \relax (t ) = c_{1}\} \\ \end{align*} \begin{align*} \{x \relax (t ) = 0\} \\ \left \{y \relax (t ) = \frac {\sqrt {-\left ({\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 t c_{1}}-1\right ) c_{1} {\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 t c_{1}}}}{{\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 t c_{1}}-1}, y \relax (t ) = -\frac {\sqrt {-\left ({\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 t c_{1}}-1\right ) c_{1} {\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 t c_{1}}}}{{\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 t c_{1}}-1}\right \} \\ \left \{z \relax (t ) = \frac {\sqrt {-\left (\frac {d}{d t}y \relax (t )\right ) y \relax (t )}}{y \relax (t )}, z \relax (t ) = -\frac {\sqrt {-\left (\frac {d}{d t}y \relax (t )\right ) y \relax (t )}}{y \relax (t )}\right \} \\ \end{align*} \begin{align*} \{x \relax (t ) = c_{1}\} \\ \{y \relax (t ) = i x \relax (t ), y \relax (t ) = -i x \relax (t )\} \\ \{z \relax (t ) = i x \relax (t ), z \relax (t ) = -i x \relax (t )\} \\ \end{align*} \begin{align*} \left \{x \relax (t ) = \frac {\sqrt {-\left ({\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 t c_{1}}-1\right ) c_{1} {\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 t c_{1}}}}{{\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 t c_{1}}-1}, x \relax (t ) = -\frac {\sqrt {-\left ({\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 t c_{1}}-1\right ) c_{1} {\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 t c_{1}}}}{{\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 t c_{1}}-1}\right \} \\ \{y \relax (t ) = 0\} \\ \left \{z \relax (t ) = \frac {\sqrt {-x \relax (t ) \left (\frac {d}{d t}x \relax (t )\right )}}{x \relax (t )}, z \relax (t ) = -\frac {\sqrt {-x \relax (t ) \left (\frac {d}{d t}x \relax (t )\right )}}{x \relax (t )}\right \} \\ \end{align*} \begin{align*} \left \{x \relax (t ) = \RootOf \left (-\left (\int _{}^{\textit {\_Z}}-\frac {2}{\sqrt {4 \textit {\_a}^{4}-16 \textit {\_a}^{2} c_{2}+16 c_{2}^{2}+c_{1}}\, \textit {\_a}}d \textit {\_a} \right )+t +c_{3}\right ), x \relax (t ) = \RootOf \left (-\left (\int _{}^{\textit {\_Z}}\frac {2}{\sqrt {4 \textit {\_a}^{4}-16 \textit {\_a}^{2} c_{2}+16 c_{2}^{2}+c_{1}}\, \textit {\_a}}d \textit {\_a} \right )+t +c_{3}\right )\right \} \\ \left \{y \relax (t ) = -\frac {\sqrt {2}\, \sqrt {x \relax (t ) \left (-x \relax (t )^{3}+\frac {d}{d t}x \relax (t )-\sqrt {x \relax (t )^{6}+2 \left (\frac {d}{d t}x \relax (t )\right )^{2}-\left (\frac {d^{2}}{d t^{2}}x \relax (t )\right ) x \relax (t )}\right )}}{2 x \relax (t )}, y \relax (t ) = \frac {\sqrt {2}\, \sqrt {x \relax (t ) \left (-x \relax (t )^{3}+\frac {d}{d t}x \relax (t )-\sqrt {x \relax (t )^{6}+2 \left (\frac {d}{d t}x \relax (t )\right )^{2}-\left (\frac {d^{2}}{d t^{2}}x \relax (t )\right ) x \relax (t )}\right )}}{2 x \relax (t )}, y \relax (t ) = -\frac {\sqrt {2}\, \sqrt {x \relax (t ) \left (-x \relax (t )^{3}+\frac {d}{d t}x \relax (t )+\sqrt {x \relax (t )^{6}+2 \left (\frac {d}{d t}x \relax (t )\right )^{2}-\left (\frac {d^{2}}{d t^{2}}x \relax (t )\right ) x \relax (t )}\right )}}{2 x \relax (t )}, y \relax (t ) = \frac {\sqrt {2}\, \sqrt {x \relax (t ) \left (-x \relax (t )^{3}+\frac {d}{d t}x \relax (t )+\sqrt {x \relax (t )^{6}+2 \left (\frac {d}{d t}x \relax (t )\right )^{2}-\left (\frac {d^{2}}{d t^{2}}x \relax (t )\right ) x \relax (t )}\right )}}{2 x \relax (t )}\right \} \\ \left \{z \relax (t ) = \frac {\sqrt {-x \relax (t ) \left (-x \relax (t ) y \relax (t )^{2}+\frac {d}{d t}x \relax (t )\right )}}{x \relax (t )}, z \relax (t ) = -\frac {\sqrt {-x \relax (t ) \left (-x \relax (t ) y \relax (t )^{2}+\frac {d}{d t}x \relax (t )\right )}}{x \relax (t )}\right \} \\ \end{align*}
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[{x'[t]==x[t]*(y[t]^2-z[t]^2),y'[t]==-y[t]*(z[t]^2+x[t]^2),z'[t]==z[t]*(x[t]^2+y[t]^2)},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
Not solved