Internal problem ID [10270]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 9, system of higher order odes
Problem number: 1938.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=\frac {f \left (t \right )}{x \left (t \right )^{2}-x \left (t \right ) y \left (t \right )-x \left (t \right ) z \left (t \right )+y \left (t \right ) z \left (t \right )}\\ y^{\prime }\left (t \right )&=-\frac {f \left (t \right )}{x \left (t \right ) y \left (t \right )-x \left (t \right ) z \left (t \right )-y \left (t \right )^{2}+y \left (t \right ) z \left (t \right )}\\ z^{\prime }\left (t \right )&=\frac {f \left (t \right )}{x \left (t \right ) y \left (t \right )-x \left (t \right ) z \left (t \right )-y \left (t \right ) z \left (t \right )+z \left (t \right )^{2}} \end {align*}
✓ Solution by Maple
Time used: 2.313 (sec). Leaf size: 1121
dsolve([(x(t)-y(t))*(x(t)-z(t))*diff(x(t),t)=f(t),(y(t)-x(t))*(y(t)-z(t))*diff(y(t),t)=f(t),(z(t)-x(t))*(z(t)-y(t))*diff(z(t),t)=f(t)],[x(t), y(t), z(t)], singsol=all)
\begin{align*} z \left (t \right ) = \int \frac {6 f \left (t \right ) \left (c_{1}^{4}+11664 \left (\int f \left (t \right )d t \right )^{2} c_{1} -23328 \left (\int f \left (t \right )d t \right ) c_{1} c_{2} +11664 c_{1} c_{2}^{2}+{\left (\left (1+108 \sqrt {\frac {\left (\int f \left (t \right )d t -c_{2} \right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \left (t \right )d t \right )+11664 \left (\int f \left (t \right )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \left (t \right )d t \right )+11664 \left (\int f \left (t \right )d t \right )^{2}\right )^{2}\right )}^{\frac {2}{3}}\right )}{{\left (\left (1+108 \sqrt {\frac {\left (\int f \left (t \right )d t -c_{2} \right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \left (t \right )d t \right )+11664 \left (\int f \left (t \right )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \left (t \right )d t \right )+11664 \left (\int f \left (t \right )d t \right )^{2}\right )^{2}\right )}^{\frac {1}{3}} \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \left (t \right )d t \right )+11664 \left (\int f \left (t \right )d t \right )^{2}\right )}d t +c_{3} z \left (t \right ) = \int \frac {3 f \left (t \right ) \left (i \sqrt {3}\, c_{1}^{4}+11664 i \sqrt {3}\, \left (\int f \left (t \right )d t \right )^{2} c_{1} -23328 i \sqrt {3}\, \left (\int f \left (t \right )d t \right ) c_{1} c_{2} +11664 i \sqrt {3}\, c_{1} c_{2}^{2}-i \sqrt {3}\, {\left (\left (1+108 \sqrt {\frac {\left (\int f \left (t \right )d t -c_{2} \right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \left (t \right )d t \right )+11664 \left (\int f \left (t \right )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \left (t \right )d t \right )+11664 \left (\int f \left (t \right )d t \right )^{2}\right )^{2}\right )}^{\frac {2}{3}}-c_{1}^{4}-11664 \left (\int f \left (t \right )d t \right )^{2} c_{1} +23328 \left (\int f \left (t \right )d t \right ) c_{1} c_{2} -11664 c_{1} c_{2}^{2}-{\left (\left (1+108 \sqrt {\frac {\left (\int f \left (t \right )d t -c_{2} \right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \left (t \right )d t \right )+11664 \left (\int f \left (t \right )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \left (t \right )d t \right )+11664 \left (\int f \left (t \right )d t \right )^{2}\right )^{2}\right )}^{\frac {2}{3}}\right )}{{\left (\left (1+108 \sqrt {\frac {\left (\int f \left (t \right )d t -c_{2} \right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \left (t \right )d t \right )+11664 \left (\int f \left (t \right )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \left (t \right )d t \right )+11664 \left (\int f \left (t \right )d t \right )^{2}\right )^{2}\right )}^{\frac {1}{3}} \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \left (t \right )d t \right )+11664 \left (\int f \left (t \right )d t \right )^{2}\right )}d t +c_{3} z \left (t \right ) = \int -\frac {3 f \left (t \right ) \left (i \sqrt {3}\, c_{1}^{4}+11664 i \sqrt {3}\, \left (\int f \left (t \right )d t \right )^{2} c_{1} -23328 i \sqrt {3}\, \left (\int f \left (t \right )d t \right ) c_{1} c_{2} +11664 i \sqrt {3}\, c_{1} c_{2}^{2}-i \sqrt {3}\, {\left (\left (1+108 \sqrt {\frac {\left (\int f \left (t \right )d t -c_{2} \right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \left (t \right )d t \right )+11664 \left (\int f \left (t \right )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \left (t \right )d t \right )+11664 \left (\int f \left (t \right )d t \right )^{2}\right )^{2}\right )}^{\frac {2}{3}}+c_{1}^{4}+11664 \left (\int f \left (t \right )d t \right )^{2} c_{1} -23328 \left (\int f \left (t \right )d t \right ) c_{1} c_{2} +11664 c_{1} c_{2}^{2}+{\left (\left (1+108 \sqrt {\frac {\left (\int f \left (t \right )d t -c_{2} \right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \left (t \right )d t \right )+11664 \left (\int f \left (t \right )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \left (t \right )d t \right )+11664 \left (\int f \left (t \right )d t \right )^{2}\right )^{2}\right )}^{\frac {2}{3}}\right )}{{\left (\left (1+108 \sqrt {\frac {\left (\int f \left (t \right )d t -c_{2} \right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \left (t \right )d t \right )+11664 \left (\int f \left (t \right )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \left (t \right )d t \right )+11664 \left (\int f \left (t \right )d t \right )^{2}\right )^{2}\right )}^{\frac {1}{3}} \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \left (t \right )d t \right )+11664 \left (\int f \left (t \right )d t \right )^{2}\right )}d t +c_{3} \end{align*} \begin{align*} y \left (t \right ) = \frac {4 \left (\frac {d}{d t}z \left (t \right )\right )^{3} z \left (t \right )+\left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right ) f \left (t \right )-\left (\frac {d}{d t}f \left (t \right )\right ) \left (\frac {d}{d t}z \left (t \right )\right )-\sqrt {-16 \left (\frac {d}{d t}z \left (t \right )\right )^{5} f \left (t \right )+\left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right )^{2} f \left (t \right )^{2}-2 \left (\frac {d}{d t}z \left (t \right )\right ) \left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right ) f \left (t \right ) \left (\frac {d}{d t}f \left (t \right )\right )+\left (\frac {d}{d t}z \left (t \right )\right )^{2} \left (\frac {d}{d t}f \left (t \right )\right )^{2}}}{4 \left (\frac {d}{d t}z \left (t \right )\right )^{3}} y \left (t \right ) = \frac {4 \left (\frac {d}{d t}z \left (t \right )\right )^{3} z \left (t \right )+\left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right ) f \left (t \right )-\left (\frac {d}{d t}f \left (t \right )\right ) \left (\frac {d}{d t}z \left (t \right )\right )+\sqrt {-16 \left (\frac {d}{d t}z \left (t \right )\right )^{5} f \left (t \right )+\left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right )^{2} f \left (t \right )^{2}-2 \left (\frac {d}{d t}z \left (t \right )\right ) \left (\frac {d^{2}}{d t^{2}}z \left (t \right )\right ) f \left (t \right ) \left (\frac {d}{d t}f \left (t \right )\right )+\left (\frac {d}{d t}z \left (t \right )\right )^{2} \left (\frac {d}{d t}f \left (t \right )\right )^{2}}}{4 \left (\frac {d}{d t}z \left (t \right )\right )^{3}} \end{align*} \begin{align*} x \left (t \right ) = \frac {z \left (t \right )^{2} \left (\frac {d}{d t}z \left (t \right )\right )-y \left (t \right ) z \left (t \right ) \left (\frac {d}{d t}z \left (t \right )\right )-f \left (t \right )}{z \left (t \right ) \left (\frac {d}{d t}z \left (t \right )\right )-\left (\frac {d}{d t}z \left (t \right )\right ) y \left (t \right )} \end{align*}
✓ Solution by Mathematica
Time used: 0.233 (sec). Leaf size: 1557
DSolve[{(x[t]-y[t])*(x[t]-z[t])*x'[t]==f[t],(y[t]-x[t])*(y[t]-z[t])*y'[t]==f[t],(z[t]-x[t])*(z[t]-y[t])*z'[t]==f[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{6} \left (2^{2/3} \sqrt [3]{27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3}+\frac {2 \sqrt [3]{2} \left (c_1{}^2-3 c_2\right )}{\sqrt [3]{27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3}}+2 c_1\right ) y(t)\to -\frac {\sqrt {-\frac {-8 c_1{}^2 \left (27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^{2/3}+2 \sqrt [3]{2} \left (27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^{4/3}+24 c_2 \left (27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^{2/3}+4\ 2^{2/3} c_1{}^4-24\ 2^{2/3} c_2 c_1{}^2+36\ 2^{2/3} c_2{}^2}{\left (27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^{2/3}}}}{4 \sqrt {3}}-\frac {\sqrt [3]{27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3}}{6 \sqrt [3]{2}}-\frac {c_1{}^2-3 c_2}{3\ 2^{2/3} \sqrt [3]{27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3}}+\frac {c_1}{3} z(t)\to \frac {4 c_1 \sqrt [3]{27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3}-2^{2/3} \left (27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^{2/3}+\sqrt {3} \sqrt [3]{27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3} \sqrt {-\frac {-8 c_1{}^2 \left (27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^{2/3}+2 \sqrt [3]{2} \left (27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^{4/3}+24 c_2 \left (27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^{2/3}+4\ 2^{2/3} c_1{}^4-24\ 2^{2/3} c_2 c_1{}^2+36\ 2^{2/3} c_2{}^2}{\left (27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^{2/3}}}-2 \sqrt [3]{2} c_1{}^2+6 \sqrt [3]{2} c_2}{12 \sqrt [3]{27 \int _1^tf(K[1])dK[1]+\sqrt {\left (27 \int _1^tf(K[1])dK[1]+2 c_1{}^3-9 c_2 c_1+27 c_3\right ){}^2-4 \left (c_1{}^2-3 c_2\right ){}^3}+2 c_1{}^3-9 c_2 c_1+27 c_3}} \end{align*}