11.23   ODE No. 1935

$$\boxed{{\displaystyle \{\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)=x\left(t\right)({\left(y\left(t\right)\right)}^2-{\left(z\left(t\right)\right)}^2),\frac{\mathrm{d}}{\mathrm{d}t}y\left(t\right)=y\left(t\right)({\left(z\left(t\right)\right)}^2-{\left(x\left(t\right)\right)}^2),\frac{\mathrm{d}}{\mathrm{d}t}z\left(t\right)=z\left(t\right)({\left(x\left(t\right)\right)}^2-{\left(y\left(t\right)\right)}^2)\}}}$$

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

Mathematica: cpu = 0.050506 (sec), leaf count = 67 $$\text{DSolve}[\{x^{\prime }(t)=x(t)(y(t{)}^2-z(t{)}^2),y^{\prime }(t)=y(t)(z(t{)}^2-x(t{)}^2),z^{\prime }(t)=z(t)(x(t{)}^2-y(t{)}^2)\},\{x(t),y(t),z(t)\},t]$$

Maple: cpu = 1.357 (sec), leaf count = 741 $$\{[\{x\left(t\right)=0\},\{y\left(t\right)=0\},\{z\left(t\right)=\mathit{\_}\mathit{C}\mathit{1}\}],[\{x\left(t\right)=0\},\{y\left(t\right)=\frac{1}{{\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{2}\mathit{\_}\mathit{C}\mathit{1}}\right)}^2{\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{1}t}\right)}^2-1}\sqrt{({\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{2}\mathit{\_}\mathit{C}\mathit{1}}\right)}^2{\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{1}t}\right)}^2-1)\mathit{\_}\mathit{C}\mathit{1}{\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{2}\mathit{\_}\mathit{C}\mathit{1}}\right)}^2{\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{1}t}\right)}^2},y\left(t\right)=-\frac{1}{{\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{2}\mathit{\_}\mathit{C}\mathit{1}}\right)}^2{\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{1}t}\right)}^2-1}\sqrt{({\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{2}\mathit{\_}\mathit{C}\mathit{1}}\right)}^2{\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{1}t}\right)}^2-1)\mathit{\_}\mathit{C}\mathit{1}{\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{2}\mathit{\_}\mathit{C}\mathit{1}}\right)}^2{\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{1}t}\right)}^2}\},\{z\left(t\right)=\frac{1}{y\left(t\right)}\sqrt{y\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}y\left(t\right)},z\left(t\right)=-\frac{1}{y\left(t\right)}\sqrt{y\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}y\left(t\right)}\}],[\{x\left(t\right)=\mathit{\_}\mathit{C}\mathit{1}\},\{y\left(t\right)=-x\left(t\right)\},\{z\left(t\right)=-x\left(t\right)\}],[\{x\left(t\right)=\mathit{\_}\mathit{C}\mathit{1}\},\{y\left(t\right)=-x\left(t\right)\},\{z\left(t\right)=x\left(t\right)\}],[\{x\left(t\right)=\mathit{\_}\mathit{C}\mathit{1}\},\{y\left(t\right)=x\left(t\right)\},\{z\left(t\right)=-x\left(t\right)\}],[\{x\left(t\right)=\mathit{\_}\mathit{C}\mathit{1}\},\{y\left(t\right)=x\left(t\right)\},\{z\left(t\right)=x\left(t\right)\}],[\{x\left(t\right)=\mathit{O}\mathit{D}\mathit{E}\mathit{S}\mathit{o}\mathit{l}\mathit{S}\mathit{t}\mathit{r}\mathit{u}\mathit{c}({\mathrm{e}}^{\int \mathit{\_}\mathit{g}\left(\mathit{\_}\mathit{f}\right)\mathrm{d}\mathit{\_}\mathit{f}+\mathit{\_}\mathit{C}\mathit{2}},[\{\frac{\mathrm{d}}{\mathrm{d}\mathit{\_}\mathit{f}}\mathit{\_}\mathit{g}\left(\mathit{\_}\mathit{f}\right)=4\frac{{\left(\mathit{\_}\mathit{g}\left(\mathit{\_}\mathit{f}\right)\right)}^3}{\mathit{\_}\mathit{f}}(3{\mathit{\_}\mathit{f}}^2+2\sqrt{\frac{3{\mathit{\_}\mathit{f}}^2\mathit{\_}\mathit{g}\left(\mathit{\_}\mathit{f}\right)+\mathit{\_}\mathit{g}\left(\mathit{\_}\mathit{f}\right)+\mathit{\_}\mathit{f}}{\mathit{\_}\mathit{g}\left(\mathit{\_}\mathit{f}\right)}}+1)+10{\left(\mathit{\_}\mathit{g}\left(\mathit{\_}\mathit{f}\right)\right)}^2+\frac{\mathit{\_}\mathit{g}\left(\mathit{\_}\mathit{f}\right)}{\mathit{\_}\mathit{f}}\},\{\mathit{\_}\mathit{f}=\frac{\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)}{{\left(x\left(t\right)\right)}^3},\mathit{\_}\mathit{g}\left(\mathit{\_}\mathit{f}\right)=-{\left(x\left(t\right)\right)}^3{(-\frac{\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}x\left(t\right)\right)x\left(t\right)}{\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)}+3\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right))}^{-1}\},\{t=\int \frac{\mathit{\_}\mathit{g}\left(\mathit{\_}\mathit{f}\right)}{\mathit{\_}\mathit{f}{\left({\mathrm{e}}^{\int \mathit{\_}\mathit{g}\left(\mathit{\_}\mathit{f}\right)\mathrm{d}\mathit{\_}\mathit{f}+\mathit{\_}\mathit{C}\mathit{2}}\right)}^2}\mathrm{d}\mathit{\_}\mathit{f}+\mathit{\_}\mathit{C}\mathit{1},x\left(t\right)={\mathrm{e}}^{\int \mathit{\_}\mathit{g}\left(\mathit{\_}\mathit{f}\right)\mathrm{d}\mathit{\_}\mathit{f}+\mathit{\_}\mathit{C}\mathit{2}}\}])\},\{y\left(t\right)=-\frac{1}{4{\left(x\left(t\right)\right)}^2\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)}\sqrt{x\left(t\right)\left(\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)\right)(4{\left(x\left(t\right)\right)}^5\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)+8{\left(x\left(t\right)\right)}^2{\left(\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)\right)}^2-\left(\frac{{\mathrm{d}}^3}{\mathrm{d}{t}^3}x\left(t\right)\right)x\left(t\right)+\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}x\left(t\right)\right)\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right))},y\left(t\right)=\frac{1}{4{\left(x\left(t\right)\right)}^2\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)}\sqrt{x\left(t\right)\left(\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)\right)(4{\left(x\left(t\right)\right)}^5\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)+8{\left(x\left(t\right)\right)}^2{\left(\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)\right)}^2-\left(\frac{{\mathrm{d}}^3}{\mathrm{d}{t}^3}x\left(t\right)\right)x\left(t\right)+\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}x\left(t\right)\right)\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right))}\},\{z\left(t\right)=\frac{1}{x\left(t\right)}\sqrt{x\left(t\right)(x\left(t\right){\left(y\left(t\right)\right)}^2-\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right))},z\left(t\right)=-\frac{1}{x\left(t\right)}\sqrt{x\left(t\right)(x\left(t\right){\left(y\left(t\right)\right)}^2-\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right))}\}],[\{x\left(t\right)=\frac{1}{{\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{2}\mathit{\_}\mathit{C}\mathit{1}}\right)}^2{\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{1}t}\right)}^2-1}\sqrt{-({\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{2}\mathit{\_}\mathit{C}\mathit{1}}\right)}^2{\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{1}t}\right)}^2-1)\mathit{\_}\mathit{C}\mathit{1}{\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{2}\mathit{\_}\mathit{C}\mathit{1}}\right)}^2{\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{1}t}\right)}^2},x\left(t\right)=-\frac{1}{{\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{2}\mathit{\_}\mathit{C}\mathit{1}}\right)}^2{\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{1}t}\right)}^2-1}\sqrt{-({\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{2}\mathit{\_}\mathit{C}\mathit{1}}\right)}^2{\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{1}t}\right)}^2-1)\mathit{\_}\mathit{C}\mathit{1}{\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{2}\mathit{\_}\mathit{C}\mathit{1}}\right)}^2{\left({\mathrm{e}}^{\mathit{\_}\mathit{C}\mathit{1}t}\right)}^2}\},\{y\left(t\right)=0\},\{z\left(t\right)=\frac{1}{x\left(t\right)}\sqrt{-x\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)},z\left(t\right)=-\frac{1}{x\left(t\right)}\sqrt{-x\left(t\right)\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)}\}]\}$$