10.10   ODE No. 1865

$$\boxed{{\displaystyle \{\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)=\mathit{a}\mathit{1}x\left(t\right)+\mathit{b}\mathit{1}y\left(t\right)+\mathit{c}\mathit{1},\frac{\mathrm{d}}{\mathrm{d}t}y\left(t\right)=\mathit{a}\mathit{2}x\left(t\right)+\mathit{b}\mathit{2}y\left(t\right)+\mathit{c}\mathit{2}\}}}$$

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

Mathematica: cpu = 1.201653 (sec), leaf count = 2062 $$\left\{\{x(t)\to -\frac{\text{b1}{e}^{-\frac{1}{2}(\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}(\frac{2((\text{a1}-\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})\text{c2}-2\text{a2}\text{c1})e^{\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}t}}{-\text{a1}-\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}}-\frac{2(2\text{a2}\text{c1}+(-\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})\text{c2})}{\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}})(e^{\frac{1}{2}(\text{a1}+\text{b2}-\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}-e^{\frac{1}{2}(\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t})}{2({\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1})}-\frac{\text{b1}{c}_2(e^{\frac{1}{2}(\text{a1}+\text{b2}-\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}-e^{\frac{1}{2}(\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t})}{\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}}+\frac{e^{-\frac{1}{2}(\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}(\frac{2(-\text{a1}\text{c1}+\text{b2}\text{c1}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}\text{c1}-2\text{b1}\text{c2})e^{\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}t}}{-\text{a1}-\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}}-\frac{2(\text{a1}\text{c1}-\text{b2}\text{c1}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}\text{c1}+2\text{b1}\text{c2})}{\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}})(-e^{\frac{1}{2}(\text{a1}+\text{b2}-\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}\text{a1}+e^{\frac{1}{2}(\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}\text{a1}+\text{b2}{e}^{\frac{1}{2}(\text{a1}+\text{b2}-\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}{e}^{\frac{1}{2}(\text{a1}+\text{b2}-\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}-\text{b2}{e}^{\frac{1}{2}(\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}{e}^{\frac{1}{2}(\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t})}{4({\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1})}+\frac{(-e^{\frac{1}{2}(\text{a1}+\text{b2}-\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}\text{a1}+e^{\frac{1}{2}(\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}\text{a1}+\text{b2}{e}^{\frac{1}{2}(\text{a1}+\text{b2}-\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}{e}^{\frac{1}{2}(\text{a1}+\text{b2}-\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}-\text{b2}{e}^{\frac{1}{2}(\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}{e}^{\frac{1}{2}(\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t})c_1}{2\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}},y(t)\to -\frac{\text{a2}{e}^{-\frac{1}{2}(\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}(\frac{2(-\text{a1}\text{c1}+\text{b2}\text{c1}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}\text{c1}-2\text{b1}\text{c2})e^{\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}t}}{-\text{a1}-\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}}-\frac{2(\text{a1}\text{c1}-\text{b2}\text{c1}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}\text{c1}+2\text{b1}\text{c2})}{\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}})(e^{\frac{1}{2}(\text{a1}+\text{b2}-\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}-e^{\frac{1}{2}(\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t})}{2({\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1})}-\frac{\text{a2}{c}_1(e^{\frac{1}{2}(\text{a1}+\text{b2}-\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}-e^{\frac{1}{2}(\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t})}{\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}}+\frac{e^{-\frac{1}{2}(\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}(\frac{2((\text{a1}-\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})\text{c2}-2\text{a2}\text{c1})e^{\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}t}}{-\text{a1}-\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}}-\frac{2(2\text{a2}\text{c1}+(-\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})\text{c2})}{\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}})(e^{\frac{1}{2}(\text{a1}+\text{b2}-\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}\text{a1}-e^{\frac{1}{2}(\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}\text{a1}-\text{b2}{e}^{\frac{1}{2}(\text{a1}+\text{b2}-\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}{e}^{\frac{1}{2}(\text{a1}+\text{b2}-\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}+\text{b2}{e}^{\frac{1}{2}(\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}{e}^{\frac{1}{2}(\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t})}{4({\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1})}+\frac{(e^{\frac{1}{2}(\text{a1}+\text{b2}-\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}\text{a1}-e^{\frac{1}{2}(\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}\text{a1}-\text{b2}{e}^{\frac{1}{2}(\text{a1}+\text{b2}-\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}{e}^{\frac{1}{2}(\text{a1}+\text{b2}-\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}+\text{b2}{e}^{\frac{1}{2}(\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}{e}^{\frac{1}{2}(\text{a1}+\text{b2}+\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}})t})c_2}{2\sqrt{{\text{a1}}^2-2\text{b2}\text{a1}+{\text{b2}}^2+4\text{a2}\text{b1}}}\}\right\}$$

Maple: cpu = 0.172 (sec), leaf count = 334 $$\left\{\{x\left(t\right)={\mathrm{e}}^{(\frac{\mathit{a}\mathit{1}}{2}+\frac{\mathit{b}\mathit{2}}{2}+\frac{1}{2}\sqrt{{\mathit{a}\mathit{1}}^2-2\mathit{a}\mathit{1}\mathit{b}\mathit{2}+4\mathit{a}\mathit{2}\mathit{b}\mathit{1}+{\mathit{b}\mathit{2}}^2})t}\mathit{\_}\mathit{C}\mathit{2}+{\mathrm{e}}^{(\frac{\mathit{a}\mathit{1}}{2}+\frac{\mathit{b}\mathit{2}}{2}-\frac{1}{2}\sqrt{{\mathit{a}\mathit{1}}^2-2\mathit{a}\mathit{1}\mathit{b}\mathit{2}+4\mathit{a}\mathit{2}\mathit{b}\mathit{1}+{\mathit{b}\mathit{2}}^2})t}\mathit{\_}\mathit{C}\mathit{1}+\frac{\mathit{c}\mathit{2}\mathit{b}\mathit{1}-\mathit{b}\mathit{2}\mathit{c}\mathit{1}}{\mathit{a}\mathit{1}\mathit{b}\mathit{2}-\mathit{a}\mathit{2}\mathit{b}\mathit{1}},y\left(t\right)=\frac{1}{(2\mathit{a}\mathit{1}\mathit{b}\mathit{2}-2\mathit{a}\mathit{2}\mathit{b}\mathit{1})\mathit{b}\mathit{1}}(-\frac{\mathit{a}\mathit{1}(2\mathit{a}\mathit{1}\mathit{b}\mathit{2}-2\mathit{a}\mathit{2}\mathit{b}\mathit{1})}{\mathit{a}\mathit{1}\mathit{b}\mathit{2}-\mathit{a}\mathit{2}\mathit{b}\mathit{1}}({\mathrm{e}}^{\frac{t}{2}(\mathit{a}\mathit{1}+\mathit{b}\mathit{2}+\sqrt{{\mathit{a}\mathit{1}}^2-2\mathit{a}\mathit{1}\mathit{b}\mathit{2}+4\mathit{a}\mathit{2}\mathit{b}\mathit{1}+{\mathit{b}\mathit{2}}^2})}\mathit{\_}\mathit{C}\mathit{2}(\mathit{a}\mathit{1}\mathit{b}\mathit{2}-\mathit{a}\mathit{2}\mathit{b}\mathit{1})+{\mathrm{e}}^{\frac{t}{2}(\mathit{a}\mathit{1}+\mathit{b}\mathit{2}-\sqrt{{\mathit{a}\mathit{1}}^2-2\mathit{a}\mathit{1}\mathit{b}\mathit{2}+4\mathit{a}\mathit{2}\mathit{b}\mathit{1}+{\mathit{b}\mathit{2}}^2})}\mathit{\_}\mathit{C}\mathit{1}(\mathit{a}\mathit{1}\mathit{b}\mathit{2}-\mathit{a}\mathit{2}\mathit{b}\mathit{1})+\mathit{c}\mathit{2}\mathit{b}\mathit{1}-\mathit{b}\mathit{2}\mathit{c}\mathit{1})+\frac{\mathit{\_}\mathit{C}\mathit{2}(2\mathit{a}\mathit{1}\mathit{b}\mathit{2}-2\mathit{a}\mathit{2}\mathit{b}\mathit{1})}{2}(\mathit{a}\mathit{1}+\mathit{b}\mathit{2}+\sqrt{{\mathit{a}\mathit{1}}^2-2\mathit{a}\mathit{1}\mathit{b}\mathit{2}+4\mathit{a}\mathit{2}\mathit{b}\mathit{1}+{\mathit{b}\mathit{2}}^2}){\mathrm{e}}^{\frac{t}{2}(\mathit{a}\mathit{1}+\mathit{b}\mathit{2}+\sqrt{{\mathit{a}\mathit{1}}^2-2\mathit{a}\mathit{1}\mathit{b}\mathit{2}+4\mathit{a}\mathit{2}\mathit{b}\mathit{1}+{\mathit{b}\mathit{2}}^2})}+\frac{\mathit{\_}\mathit{C}\mathit{1}(2\mathit{a}\mathit{1}\mathit{b}\mathit{2}-2\mathit{a}\mathit{2}\mathit{b}\mathit{1})}{2}(\mathit{a}\mathit{1}+\mathit{b}\mathit{2}-\sqrt{{\mathit{a}\mathit{1}}^2-2\mathit{a}\mathit{1}\mathit{b}\mathit{2}+4\mathit{a}\mathit{2}\mathit{b}\mathit{1}+{\mathit{b}\mathit{2}}^2}){\mathrm{e}}^{\frac{t}{2}(\mathit{a}\mathit{1}+\mathit{b}\mathit{2}-\sqrt{{\mathit{a}\mathit{1}}^2-2\mathit{a}\mathit{1}\mathit{b}\mathit{2}+4\mathit{a}\mathit{2}\mathit{b}\mathit{1}+{\mathit{b}\mathit{2}}^2})}-\mathit{c}\mathit{1}(2\mathit{a}\mathit{1}\mathit{b}\mathit{2}-2\mathit{a}\mathit{2}\mathit{b}\mathit{1}))\}\right\}$$