10.26 problem 1938

Internal problem ID [9517]

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*} \left (x \relax (t )-y \relax (t )\right ) \left (x \relax (t )-z \relax (t )\right ) x^{\prime }\relax (t )&=f \relax (t )\\ \left (y \relax (t )-x \relax (t )\right ) \left (y \relax (t )-z \relax (t )\right ) y^{\prime }\relax (t )&=f \relax (t )\\ \left (z \relax (t )-x \relax (t )\right ) \left (-y \relax (t )+z \relax (t )\right ) z^{\prime }\relax (t )&=f \relax (t ) \end {align*}

✓ Solution by Maple

Time used: 1.236 (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*} x \relax (t ) = \int \frac {6 f \relax (t ) \left (c_{1}^{4}+11664 \left (\int f \relax (t )d t \right )^{2} c_{1}-23328 \left (\int f \relax (t )d t \right ) c_{1} c_{2}+11664 c_{1} c_{2}^{2}+\left (\left (1+108 \sqrt {\frac {\left (\int f \relax (t )d t -c_{2}\right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right )^{2}\right )^{\frac {2}{3}}\right )}{\left (\left (1+108 \sqrt {\frac {\left (\int f \relax (t )d t -c_{2}\right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right )^{2}\right )^{\frac {1}{3}} \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right )}d t +c_{3} \\ x \relax (t ) = \int \frac {3 f \relax (t ) \left (-i \sqrt {3}\, c_{1}^{4}-11664 i \sqrt {3}\, \left (\int f \relax (t )d t \right )^{2} c_{1}+23328 i \sqrt {3}\, \left (\int f \relax (t )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 \relax (t )d t -c_{2}\right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right )^{2}\right )^{\frac {2}{3}}-c_{1}^{4}-11664 \left (\int f \relax (t )d t \right )^{2} c_{1}+23328 \left (\int f \relax (t )d t \right ) c_{1} c_{2}-11664 c_{1} c_{2}^{2}-\left (\left (1+108 \sqrt {\frac {\left (\int f \relax (t )d t -c_{2}\right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right )^{2}\right )^{\frac {2}{3}}\right )}{\left (\left (1+108 \sqrt {\frac {\left (\int f \relax (t )d t -c_{2}\right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right )^{2}\right )^{\frac {1}{3}} \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right )}d t +c_{3} \\ x \relax (t ) = \int -\frac {3 f \relax (t ) \left (-i \sqrt {3}\, c_{1}^{4}-11664 i \sqrt {3}\, \left (\int f \relax (t )d t \right )^{2} c_{1}+23328 i \sqrt {3}\, \left (\int f \relax (t )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 \relax (t )d t -c_{2}\right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right )^{2}\right )^{\frac {2}{3}}+c_{1}^{4}+11664 \left (\int f \relax (t )d t \right )^{2} c_{1}-23328 \left (\int f \relax (t )d t \right ) c_{1} c_{2}+11664 c_{1} c_{2}^{2}+\left (\left (1+108 \sqrt {\frac {\left (\int f \relax (t )d t -c_{2}\right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right )^{2}\right )^{\frac {2}{3}}\right )}{\left (\left (1+108 \sqrt {\frac {\left (\int f \relax (t )d t -c_{2}\right )^{2}}{c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}}}\right ) \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right )^{2}\right )^{\frac {1}{3}} \left (c_{1}^{3}+11664 c_{2}^{2}-23328 c_{2} \left (\int f \relax (t )d t \right )+11664 \left (\int f \relax (t )d t \right )^{2}\right )}d t +c_{3} \\ \end{align*} \begin{align*} y \relax (t ) = \frac {4 \left (\frac {d}{d t}x \relax (t )\right )^{3} x \relax (t )+\left (\frac {d^{2}}{d t^{2}}x \relax (t )\right ) f \relax (t )-\left (\frac {d}{d t}x \relax (t )\right ) \left (\frac {d}{d t}f \relax (t )\right )-\sqrt {-16 \left (\frac {d}{d t}x \relax (t )\right )^{5} f \relax (t )+\left (\frac {d^{2}}{d t^{2}}x \relax (t )\right )^{2} f \relax (t )^{2}-2 \left (\frac {d}{d t}x \relax (t )\right ) \left (\frac {d^{2}}{d t^{2}}x \relax (t )\right ) f \relax (t ) \left (\frac {d}{d t}f \relax (t )\right )+\left (\frac {d}{d t}x \relax (t )\right )^{2} \left (\frac {d}{d t}f \relax (t )\right )^{2}}}{4 \left (\frac {d}{d t}x \relax (t )\right )^{3}} \\ y \relax (t ) = \frac {4 \left (\frac {d}{d t}x \relax (t )\right )^{3} x \relax (t )+\left (\frac {d^{2}}{d t^{2}}x \relax (t )\right ) f \relax (t )-\left (\frac {d}{d t}x \relax (t )\right ) \left (\frac {d}{d t}f \relax (t )\right )+\sqrt {-16 \left (\frac {d}{d t}x \relax (t )\right )^{5} f \relax (t )+\left (\frac {d^{2}}{d t^{2}}x \relax (t )\right )^{2} f \relax (t )^{2}-2 \left (\frac {d}{d t}x \relax (t )\right ) \left (\frac {d^{2}}{d t^{2}}x \relax (t )\right ) f \relax (t ) \left (\frac {d}{d t}f \relax (t )\right )+\left (\frac {d}{d t}x \relax (t )\right )^{2} \left (\frac {d}{d t}f \relax (t )\right )^{2}}}{4 \left (\frac {d}{d t}x \relax (t )\right )^{3}} \\ \end{align*} \begin{align*} z \relax (t ) = \frac {-\left (\frac {d}{d t}x \relax (t )\right ) y \relax (t ) x \relax (t )+x \relax (t )^{2} \left (\frac {d}{d t}x \relax (t )\right )-f \relax (t )}{-\left (\frac {d}{d t}x \relax (t )\right ) y \relax (t )+x \relax (t ) \left (\frac {d}{d t}x \relax (t )\right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.187 (sec). Leaf size: 1536

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 \left (\int _1^tf(K[1])dK[1]+c_3\right )+\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}+\frac {2 \sqrt [3]{2} \left (c_1{}^2-3 c_2\right )}{\sqrt [3]{27 \left (\int _1^tf(K[1])dK[1]+c_3\right )+\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}}+2 c_1\right ) \\ y(t)\to \frac {1}{12} \left (-\sqrt {6} \sqrt {-\frac {-4 c_1{}^2 \left (27 \left (\int _1^tf(K[1])dK[1]+c_3\right )+\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\right ){}^{2/3}+\sqrt [3]{2} \left (27 \left (\int _1^tf(K[1])dK[1]+c_3\right )+\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\right ){}^{4/3}+12 c_2 \left (27 \left (\int _1^tf(K[1])dK[1]+c_3\right )+\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\right ){}^{2/3}+2\ 2^{2/3} c_1{}^4-12\ 2^{2/3} c_2 c_1{}^2+18\ 2^{2/3} c_2{}^2}{\left (27 \left (\int _1^tf(K[1])dK[1]+c_3\right )+\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\right ){}^{2/3}}}-2^{2/3} \sqrt [3]{27 \left (\int _1^tf(K[1])dK[1]+c_3\right )+\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}-\frac {2 \sqrt [3]{2} \left (c_1{}^2-3 c_2\right )}{\sqrt [3]{27 \left (\int _1^tf(K[1])dK[1]+c_3\right )+\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}}+4 c_1\right ) \\ z(t)\to \frac {4 c_1 \sqrt [3]{27 \left (\int _1^tf(K[1])dK[1]+c_3\right )+\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}-2^{2/3} \left (27 \left (\int _1^tf(K[1])dK[1]+c_3\right )+\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\right ){}^{2/3}+\sqrt {3} \sqrt [3]{27 \left (\int _1^tf(K[1])dK[1]+c_3\right )+\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} \sqrt {-\frac {-8 c_1{}^2 \left (27 \left (\int _1^tf(K[1])dK[1]+c_3\right )+\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\right ){}^{2/3}+2 \sqrt [3]{2} \left (27 \left (\int _1^tf(K[1])dK[1]+c_3\right )+\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\right ){}^{4/3}+24 c_2 \left (27 \left (\int _1^tf(K[1])dK[1]+c_3\right )+\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\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 \left (\int _1^tf(K[1])dK[1]+c_3\right )+\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\right ){}^{2/3}}}-2 \sqrt [3]{2} c_1{}^2+6 \sqrt [3]{2} c_2}{12 \sqrt [3]{27 \left (\int _1^tf(K[1])dK[1]+c_3\right )+\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}} \\ \end{align*}