2.1905   ODE No. 1905

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

$$\{x^{\prime }(t)=h(t)y(t)-g(t)z(t),y^{\prime }(t)=f(t)z(t)-h(t)x(t),z^{\prime }(t)=g(t)x(t)-f(t)y(t)\}$$ ✗ Mathematica : cpu = 0.00716412 (sec), leaf count = 0 , could not solve

DSolve[{Derivative[1][x][t] == h[t]*y[t] - g[t]*z[t], Derivative[1][y][t] == -(h[t]*x[t]) + f[t]*z[t], Derivative[1][z][t] == g[t]*x[t] - f[t]*y[t]}, {x[t], y[t], z[t]}, t]

✗ Maple : cpu = 0. (sec), leaf count = 0 , result contains DESol

$$\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(h\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(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)}+\frac{\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}g\left(t\right)\right)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{g\left(t\right)\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}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(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\right)f\left(t\right)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{g\left(t\right)f\left(t\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{{\mathrm{d}}^2}{\mathrm{d}{t}^2}\mathit{\_}\mathit{Y}\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(\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)}-\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)}+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)}+2\frac{{\left(\frac{\mathrm{d}}{\mathrm{d}t}g\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}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)}+\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(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)}^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{{\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)}+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)}+\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)}+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)}-3\frac{h\left(t\right)\left(\frac{\mathrm{d}}{\mathrm{d}t}g\left(t\right)\right){\left(f\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{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{h\left(t\right)\left(\frac{\mathrm{d}}{\mathrm{d}t}h\left(t\right)\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{\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}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)}-\frac{\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}h\left(t\right)\right)h\left(t\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{\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(\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}g\left(t\right)\right){\left(h\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(\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)}^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}}{\mathrm{d}t}h\left(t\right)\right)f\left(t\right)h\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(h\left(t\right)\right)}^2\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)}+\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)}+\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)}+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)}-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)}+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)}+\frac{\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}g\left(t\right)\right)h\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}}^2}{\mathrm{d}{t}^2}h\left(t\right)\right){\left(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)}-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)})\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)f\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}x\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-f\left(t\right)\left(\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)\right)g\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\}$$