10.50   ODE No. 1905

$$\boxed{{\displaystyle \{\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)=h\left(t\right)y\left(t\right)-g\left(t\right)z\left(t\right),\frac{\mathrm{d}}{\mathrm{d}t}y\left(t\right)=f\left(t\right)z\left(t\right)-h\left(t\right)x\left(t\right),\frac{\mathrm{d}}{\mathrm{d}t}z\left(t\right)=x\left(t\right)g\left(t\right)-y\left(t\right)f\left(t\right)\}}}$$

1. Problem in Latex
2. Mathematica input
3. Maple input

Mathematica: cpu = 0.006501 (sec), leaf count = 61 $$\text{DSolve}[\{x^{\prime }(t)=h(t)y(t)-g(t)z(t),y^{\prime }(t)=f(t)z(t)-h(t)x(t),z^{\prime }(t)=g(t)x(t)-f(t)y(t)\},\{x(t),y(t),z(t)\},t]$$

Maple: cpu = 0.530 (sec), leaf count = 2014 $$\left\{\{x\left(t\right)=\mathit{D}\mathit{E}\mathit{S}\mathit{o}\mathit{l}(\{\frac{{\mathrm{d}}^3}{\mathrm{d}{t}^3}\mathit{\_}\mathit{Y}\left(t\right)+(\frac{\left(\frac{\mathrm{d}}{\mathrm{d}t}f\left(t\right)\right){\left(g\left(t\right)\right)}^2}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}+\frac{{\left(h\left(t\right)\right)}^2\frac{\mathrm{d}}{\mathrm{d}t}f\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}+\frac{\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}h\left(t\right)\right)g\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-\frac{h\left(t\right)\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}g\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}+2\frac{\left(\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)\right)f\left(t\right)g\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}+2\frac{h\left(t\right)\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)f\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)})\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}\mathit{\_}\mathit{Y}\left(t\right)+(-\frac{{\left(f\left(t\right)\right)}^3{\left(g\left(t\right)\right)}^2}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-\frac{{\left(h\left(t\right)\right)}^2{\left(f\left(t\right)\right)}^3}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-\frac{f\left(t\right){\left(g\left(t\right)\right)}^4}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-\frac{{\left(h\left(t\right)\right)}^{4}f\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}+\frac{{\left(h\left(t\right)\right)}^3\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-\frac{\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right){\left(g\left(t\right)\right)}^3}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-2\frac{f\left(t\right){\left(\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)\right)}^2}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-2\frac{{\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)}^{2}f\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-\frac{\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}h\left(t\right)\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}+\frac{\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}g\left(t\right)\right)\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-2\frac{{\left(h\left(t\right)\right)}^{2}f\left(t\right){\left(g\left(t\right)\right)}^2}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}+3\frac{h\left(t\right){\left(f\left(t\right)\right)}^2\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-3\frac{\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right){\left(f\left(t\right)\right)}^{2}g\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}+\frac{h\left(t\right)\left(\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)\right){\left(g\left(t\right)\right)}^2}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-\frac{{\left(h\left(t\right)\right)}^2\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}+\frac{f\left(t\right)g\left(t\right)\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}g\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}+\frac{h\left(t\right)\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}h\left(t\right)\right)f\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-\frac{\left(\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)\right)g\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}f\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-\frac{h\left(t\right)\left(\frac{\mathrm{d}}{\mathrm{d}t}f\left(t\right)\right)\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)})\frac{\mathrm{d}}{\mathrm{d}t}\mathit{\_}\mathit{Y}\left(t\right)+(\frac{{\left(g\left(t\right)\right)}^4\frac{\mathrm{d}}{\mathrm{d}t}f\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}+\frac{{\left(h\left(t\right)\right)}^4\frac{\mathrm{d}}{\mathrm{d}t}f\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}+\frac{\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}h\left(t\right)\right){\left(g\left(t\right)\right)}^3}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-\frac{{\left(h\left(t\right)\right)}^3\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}g\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-\frac{{\left(h\left(t\right)\right)}^{2}f\left(t\right)\left(\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)\right)g\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-\frac{h\left(t\right)\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)f\left(t\right){\left(g\left(t\right)\right)}^2}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-\frac{\left(\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)\right)f\left(t\right){\left(g\left(t\right)\right)}^3}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-\frac{{\left(h\left(t\right)\right)}^3\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)f\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}+2\frac{{\left(h\left(t\right)\right)}^2{\left(g\left(t\right)\right)}^2\frac{\mathrm{d}}{\mathrm{d}t}f\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}+3\frac{h\left(t\right){\left(\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)\right)}^{2}g\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-3\frac{\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)\left(\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)\right){\left(g\left(t\right)\right)}^2}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}+3\frac{{\left(h\left(t\right)\right)}^2\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-\frac{h\left(t\right){\left(g\left(t\right)\right)}^2\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}g\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}+\frac{{\left(h\left(t\right)\right)}^2\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}h\left(t\right)\right)g\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}-3\frac{h\left(t\right){\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)}^{2}g\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)})\mathit{\_}\mathit{Y}\left(t\right)\},\left\{\mathit{\_}\mathit{Y}\left(t\right)\right\}),y\left(t\right)=\frac{-{\left(h\left(t\right)\right)}^{2}x\left(t\right)g\left(t\right)-x\left(t\right){\left(g\left(t\right)\right)}^3-h\left(t\right)\left(\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)\right)f\left(t\right)-\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}x\left(t\right)\right)g\left(t\right)+\left(\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)\right)\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)},z\left(t\right)=\frac{-{\left(h\left(t\right)\right)}^{3}x\left(t\right)-h\left(t\right)x\left(t\right){\left(g\left(t\right)\right)}^2+\left(\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)\right)g\left(t\right)f\left(t\right)-h\left(t\right)\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}x\left(t\right)+\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)}{-{\left(h\left(t\right)\right)}^{2}f\left(t\right)-f\left(t\right){\left(g\left(t\right)\right)}^2+h\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)-\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)g\left(t\right)}\}\right\}$$