Internal problem ID [9511]
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 }\left (t \right )&=x \left (t \right ) \left (y \left (t \right )^{2}-z \left (t \right )^{2}\right )\\ y^{\prime }\left (t \right )&=-y \left (t \right ) \left (z \left (t \right )^{2}+x \left (t \right )^{2}\right )\\ z^{\prime }\left (t \right )&=z \left (t \right ) \left (x \left (t \right )^{2}+y \left (t \right )^{2}\right ) \end {align*}
✓ Solution by Maple
Time used: 1.156 (sec). Leaf size: 480
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*} \{z \left (t \right ) = 0\} \\ \{y \left (t \right ) = 0\} \\ \{x \left (t \right ) = c_{1}\} \\ \end{align*} \begin{align*} \{z \left (t \right ) = 0\} \\ \left \{y \left (t \right ) = \frac {\sqrt {-\left ({\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}-1\right ) c_{1} {\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}}}{{\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}-1}, y \left (t \right ) = -\frac {\sqrt {-\left ({\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}-1\right ) c_{1} {\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}}}{{\mathrm e}^{2 c_{2} c_{1}} {\mathrm e}^{2 c_{1} t}-1}\right \} \\ \left \{x \left (t \right ) = \frac {\sqrt {-y \left (t \right ) \left (\frac {d}{d t}y \left (t \right )\right )}}{y \left (t \right )}, x \left (t \right ) = -\frac {\sqrt {-y \left (t \right ) \left (\frac {d}{d t}y \left (t \right )\right )}}{y \left (t \right )}\right \} \\ \end{align*} \begin{align*} \left \{z \left (t \right ) = \frac {\sqrt {\left ({\mathrm e}^{2 c_{3} c_{2}} {\mathrm e}^{2 c_{2} t}-1\right ) c_{2} {\mathrm e}^{2 c_{3} c_{2}} {\mathrm e}^{2 c_{2} t}}}{{\mathrm e}^{2 c_{3} c_{2}} {\mathrm e}^{2 c_{2} t}-1}, z \left (t \right ) = -\frac {\sqrt {\left ({\mathrm e}^{2 c_{3} c_{2}} {\mathrm e}^{2 c_{2} t}-1\right ) c_{2} {\mathrm e}^{2 c_{3} c_{2}} {\mathrm e}^{2 c_{2} t}}}{{\mathrm e}^{2 c_{3} c_{2}} {\mathrm e}^{2 c_{2} t}-1}\right \} \\ \left \{y \left (t \right ) = \frac {\sqrt {\frac {{\mathrm e}^{-2 \left (\int z \left (t \right )^{2}d t \right )} \left (c_{1} -2 \left (\int {\mathrm e}^{-2 \left (\int \frac {z \left (t \right )^{3}+\frac {d}{d t}z \left (t \right )}{z \left (t \right )}d t \right )}d t \right )\right )}{z \left (t \right )^{2}}}}{c_{1} -2 \left (\int {\mathrm e}^{-2 \left (\int \frac {z \left (t \right )^{3}+\frac {d}{d t}z \left (t \right )}{z \left (t \right )}d t \right )}d t \right )}, y \left (t \right ) = -\frac {\sqrt {\frac {{\mathrm e}^{-2 \left (\int z \left (t \right )^{2}d t \right )} \left (c_{1} -2 \left (\int {\mathrm e}^{-2 \left (\int \frac {z \left (t \right )^{3}+\frac {d}{d t}z \left (t \right )}{z \left (t \right )}d t \right )}d t \right )\right )}{z \left (t \right )^{2}}}}{c_{1} -2 \left (\int {\mathrm e}^{-2 \left (\int \frac {z \left (t \right )^{3}+\frac {d}{d t}z \left (t \right )}{z \left (t \right )}d t \right )}d t \right )}\right \} \\ \left \{x \left (t \right ) = \frac {\sqrt {-z \left (t \right ) \left (z \left (t \right ) y \left (t \right )^{2}-\frac {d}{d t}z \left (t \right )\right )}}{z \left (t \right )}, x \left (t \right ) = -\frac {\sqrt {-z \left (t \right ) \left (z \left (t \right ) y \left (t \right )^{2}-\frac {d}{d t}z \left (t \right )\right )}}{z \left (t \right )}\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