Internal problem ID [10245]
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: 0.172 (sec). Leaf size: 53
dsolve([diff(x(t),t)=-x(t)*(x(t)+y(t)),diff(y(t),t)=y(t)*(x(t)+y(t))],[x(t), y(t)], singsol=all)
\begin{align*} \{y \left (t \right ) = 0\} \left \{x \left (t \right ) = \frac {1}{t +c_{1}}\right \} \end{align*} \begin{align*} \left \{y \left (t \right ) = \frac {\tan \left (\frac {c_{2} +t}{c_{1}}\right )}{c_{1}}\right \} \left \{x \left (t \right ) = \frac {-y \left (t \right )^{2}+\frac {d}{d t}y \left (t \right )}{y \left (t \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*}