Internal problem ID [10236]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 9, system of higher order odes
Problem number: 1913.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )^{2}-x \left (t \right ) y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right ) y \left (t \right )+y \left (t \right )^{2} \end {align*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 54
dsolve([diff(x(t),t)=-x(t)*(x(t)+y(t)),diff(y(t),t)=y(t)*(x(t)+y(t))],singsol=all)
\begin{align*} \left [\{x \left (t \right ) = 0\}, \left \{y \left (t \right ) &= \frac {1}{-t +c_{1}}\right \}\right ] \\ \left [\left \{x \left (t \right ) &= \frac {\tanh \left (\frac {c_{2} +t}{c_{1}}\right )}{c_{1}}\right \}, \left \{y \left (t \right ) &= -\frac {x \left (t \right )^{2}+\frac {d}{d t}x \left (t \right )}{x \left (t \right )}\right \}\right ] \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.029 (sec). Leaf size: 52
DSolve[{x'[t]==-x[t]*(x[t]+y[t]),y'[t]==y[t]*(x[t]+y[t])},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to -\sqrt {c_1} \cot \left (\sqrt {c_1} (t-c_2)\right ) \\ x(t)\to -\sqrt {c_1} \tan \left (\sqrt {c_1} (t-c_2)\right ) \\ \end{align*}