Internal problem ID [9492]
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 }\relax (t )&=-x \relax (t ) \left (x \relax (t )+y \relax (t )\right )\\ y^{\prime }\relax (t )&=y \relax (t ) \left (x \relax (t )+y \relax (t )\right ) \end {align*}
✓ Solution by Maple
Time used: 0.109 (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))},{x(t), y(t)}, singsol=all)
\begin{align*} \{x \relax (t ) = 0\} \\ \left \{y \relax (t ) = \frac {1}{c_{1}-t}\right \} \\ \end{align*} \begin{align*} \left \{x \relax (t ) = \frac {\tanh \left (\frac {c_{2}+t}{c_{1}}\right )}{c_{1}}\right \} \\ \left \{y \relax (t ) = -\frac {x \relax (t )^{2}+\frac {d}{d t}x \relax (t )}{x \relax (t )}\right \} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.022 (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*}