Internal problem ID [9495]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 9, system of higher order odes
Problem number: 1916.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=h \left (a -x \relax (t )\right ) \left (c -x \relax (t )-y \relax (t )\right )\\ y^{\prime }\relax (t )&=k \left (b -y \relax (t )\right ) \left (c -x \relax (t )-y \relax (t )\right ) \end {align*}
✓ Solution by Maple
Time used: 0.356 (sec). Leaf size: 237
dsolve({diff(x(t),t)=h*(a-x(t))*(c-x(t)-y(t)),diff(y(t),t)=k*(b-y(t))*(c-x(t)-y(t))},{x(t), y(t)}, singsol=all)
\begin{align*} \{x \relax (t ) = a\} \\ \left \{y \relax (t ) = -\frac {a \,{\mathrm e}^{a c_{1} k +a k t +b c_{1} k +b k t -c c_{1} k -c k t}-c \,{\mathrm e}^{a c_{1} k +a k t +b c_{1} k +b k t -c c_{1} k -c k t}+b}{-1+{\mathrm e}^{a c_{1} k +a k t +b c_{1} k +b k t -c c_{1} k -c k t}}\right \} \\ \end{align*} \begin{align*} \left \{x \relax (t ) = \RootOf \left (-\left (\int _{}^{\textit {\_Z}}\frac {\left (\textit {\_a} -a \right )^{-\frac {k}{h}}}{\left (\left (\textit {\_a} -a \right )^{-\frac {k}{h}} h \textit {\_a} +\left (\textit {\_a} -a \right )^{-\frac {k}{h}} h b -\left (\textit {\_a} -a \right )^{-\frac {k}{h}} h c +c_{1}\right ) \left (\textit {\_a} -a \right )}d \textit {\_a} \right )+t +c_{2}\right )\right \} \\ \left \{y \relax (t ) = \frac {x \relax (t ) a h -a c h -x \relax (t )^{2} h +x \relax (t ) c h +\frac {d}{d t}x \relax (t )}{h x \relax (t )-a h}\right \} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.186 (sec). Leaf size: 277
DSolve[{x'[t]==h*(a-x[t])*(c-x[t]-y[t]),y'[t]==k*(b-y[t])*(c-x[t]-y[t])},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to b+c_1 \left (h \left (a-\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {(h (a-K[1]))^{\frac {k}{h}}}{(a-K[1]) \left (c_1 (a h-h K[1])^{\frac {k}{h}} (h (a-K[1]))^{\frac {k}{h}}-c (h (a-K[1]))^{\frac {k}{h}}+K[1] (h (a-K[1]))^{\frac {k}{h}}+b (a h-h K[1])^{\frac {k}{h}}\right )}dK[1]\&\right ][-h t+c_2]\right )\right ){}^{\frac {k}{h}} \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {(h (a-K[1]))^{\frac {k}{h}}}{(a-K[1]) \left (c_1 (a h-h K[1])^{\frac {k}{h}} (h (a-K[1]))^{\frac {k}{h}}-c (h (a-K[1]))^{\frac {k}{h}}+K[1] (h (a-K[1]))^{\frac {k}{h}}+b (a h-h K[1])^{\frac {k}{h}}\right )}dK[1]\&\right ][-h t+c_2] \\ \end{align*}