Internal problem ID [9508]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 9, system of higher order odes
Problem number: 1933.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=x \left (t \right ) y \left (t \right )\\ x^{\prime }\left (t \right )+z^{\prime }\left (t \right )&=x \left (t \right ) z \left (t \right )\\ y^{\prime }\left (t \right )+z^{\prime }\left (t \right )&=y \left (t \right ) z \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 1.5 (sec). Leaf size: 4316
dsolve([diff(x(t),t)+diff(y(t),t)=x(t)*y(t),diff(y(t),t)+diff(z(t),t)=y(t)*z(t),diff(x(t),t)+diff(z(t),t)=x(t)*z(t)],[x(t), y(t), z(t)], singsol=all)
\begin{align*} \left \{z \left (t \right ) = \frac {2}{2 c_{2} -t}\right \} \\ \left \{y \left (t \right ) = \left (\int -\frac {z \left (t \right )^{2} {\mathrm e}^{-\left (\int z \left (t \right )d t \right )}}{2}d t +c_{1} \right ) {\mathrm e}^{\int z \left (t \right )d t}\right \} \\ \{x \left (t \right ) = z \left (t \right )\} \\ \end{align*} \begin{align*} \left \{z \left (t \right ) = \frac {2}{2 c_{2} -t}\right \} \\ \{y \left (t \right ) = z \left (t \right )\} \\ \left \{x \left (t \right ) = \left (\int -\frac {z \left (t \right )^{2} {\mathrm e}^{-\left (\int z \left (t \right )d t \right )}}{2}d t +c_{1} \right ) {\mathrm e}^{\int z \left (t \right )d t}\right \} \\ \end{align*} \begin{align*} \text {Expression too large to display} \\ \left \{y \left (t \right ) = \frac {-z \left (t \right ) \left (\frac {d}{d t}z \left (t \right )\right )+\frac {d^{2}}{d t^{2}}z \left (t \right )-\sqrt {2 z \left (t \right )^{3} \left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right )-3 z \left (t \right )^{2} \left (\frac {d}{d t}z \left (t \right )\right )^{2}-6 \left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right ) \left (\frac {d}{d t}z \left (t \right )\right ) z \left (t \right )+8 \left (\frac {d}{d t}z \left (t \right )\right )^{3}+\left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right )^{2}}}{-z \left (t \right )^{2}+2 \frac {d}{d t}z \left (t \right )}, y \left (t \right ) = \frac {-z \left (t \right ) \left (\frac {d}{d t}z \left (t \right )\right )+\frac {d^{2}}{d t^{2}}z \left (t \right )+\sqrt {2 z \left (t \right )^{3} \left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right )-3 z \left (t \right )^{2} \left (\frac {d}{d t}z \left (t \right )\right )^{2}-6 \left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right ) \left (\frac {d}{d t}z \left (t \right )\right ) z \left (t \right )+8 \left (\frac {d}{d t}z \left (t \right )\right )^{3}+\left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right )^{2}}}{-z \left (t \right )^{2}+2 \frac {d}{d t}z \left (t \right )}\right \} \\ \left \{x \left (t \right ) = \frac {z \left (t \right ) y \left (t \right )-2 \frac {d}{d t}z \left (t \right )}{y \left (t \right )-z \left (t \right )}\right \} \\ \end{align*}
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[{x'[t]+y'[t]==x[t]*y[t],y'[t]+z'[t]==y[t]*z[t],x'[t]+z'[t]==x[t]*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
Not solved