Internal problem ID [10239]
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 }\left (t \right )&=x \left (t \right )^{2} h +x \left (t \right ) y \left (t \right ) h -x \left (t \right ) a h -x \left (t \right ) c h -y \left (t \right ) a h +a c h\\ y^{\prime }\left (t \right )&=x \left (t \right ) y \left (t \right ) k -x \left (t \right ) b k +y \left (t \right )^{2} k -y \left (t \right ) b k -y \left (t \right ) c k +b c k \end {align*}
✓ Solution by Maple
Time used: 0.469 (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))],singsol=all)
\begin{align*} \left [\{x \left (t \right ) = a\}, \left \{y \left (t \right ) &= -\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 \}\right ] \\ \left [\left \{x \left (t \right ) &= \operatorname {RootOf}\left (-\left (\int _{}^{\textit {\_Z}}\frac {\left (\textit {\_a} -a \right )^{-\frac {k}{h}}}{\left (h \left (\textit {\_a} -a \right )^{-\frac {k}{h}} \textit {\_a} +h \left (\textit {\_a} -a \right )^{-\frac {k}{h}} b -h \left (\textit {\_a} -a \right )^{-\frac {k}{h}} c +c_{1} \right ) \left (\textit {\_a} -a \right )}d \textit {\_a} \right )+t +c_{2} \right )\right \}, \left \{y \left (t \right ) &= \frac {-x \left (t \right )^{2} h +x \left (t \right ) a h +x \left (t \right ) c h -a c h +\frac {d}{d t}x \left (t \right )}{x \left (t \right ) h -h a}\right \}\right ] \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.764 (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*}