\[ \left \{-a y'(t)+b x(t)+x''(t)=0,a x'(t)+b y(t)+y''(t)=0\right \} \] ✓ Mathematica : cpu = 0.239395 (sec), leaf count = 4815
\[\left \{\left \{x(t)\to \frac {e^{-\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}-\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} \left (e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} a^2-e^{\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} a^2-e^{\sqrt {2} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t+\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} a^2+e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}+\sqrt {2} \sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t} a^2+\sqrt {a^2 \left (a^2+4 b\right )} e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}+\sqrt {a^2 \left (a^2+4 b\right )} e^{\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}+\sqrt {a^2 \left (a^2+4 b\right )} e^{\sqrt {2} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t+\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}+\sqrt {a^2 \left (a^2+4 b\right )} e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}+\sqrt {2} \sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}\right ) c_1}{4 \sqrt {a^2 \left (a^2+4 b\right )}}+\frac {e^{-\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}-\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} \left (\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} a^2-\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} e^{\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} a^2+\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} e^{\sqrt {2} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t+\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} a^2-\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}+\sqrt {2} \sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t} a^2-\sqrt {a^2 \left (a^2+4 b\right )} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}-\sqrt {a^2 \left (a^2+4 b\right )} \sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} e^{\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}+\sqrt {a^2 \left (a^2+4 b\right )} \sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} e^{\sqrt {2} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t+\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}+\sqrt {a^2 \left (a^2+4 b\right )} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}+\sqrt {2} \sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}\right ) c_2}{2 \sqrt {2} \sqrt {a^2 \left (a^2+4 b\right )} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} \sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}}}-\frac {a b e^{-\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}-\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} \left (-e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}}+e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}+\sqrt {2} \sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}}+\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} e^{\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}-\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} e^{\sqrt {2} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t+\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}\right ) c_3}{\sqrt {2} \sqrt {a^2 \left (a^2+4 b\right )} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} \sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}}}-\frac {a e^{-\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}-\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} \left (-e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}+e^{\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}+e^{\sqrt {2} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t+\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}-e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}+\sqrt {2} \sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}\right ) c_4}{2 \sqrt {a^2 \left (a^2+4 b\right )}},y(t)\to \frac {a b e^{-\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}-\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} \left (-e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}}+e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}+\sqrt {2} \sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}}+\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} e^{\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}-\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} e^{\sqrt {2} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t+\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}\right ) c_1}{\sqrt {2} \sqrt {a^2 \left (a^2+4 b\right )} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} \sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}}}+\frac {a e^{-\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}-\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} \left (-e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}+e^{\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}+e^{\sqrt {2} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t+\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}-e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}+\sqrt {2} \sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}\right ) c_2}{2 \sqrt {a^2 \left (a^2+4 b\right )}}+\frac {e^{-\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}-\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} \left (e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} a^2-e^{\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} a^2-e^{\sqrt {2} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t+\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} a^2+e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}+\sqrt {2} \sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t} a^2+\sqrt {a^2 \left (a^2+4 b\right )} e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}+\sqrt {a^2 \left (a^2+4 b\right )} e^{\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}+\sqrt {a^2 \left (a^2+4 b\right )} e^{\sqrt {2} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t+\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}+\sqrt {a^2 \left (a^2+4 b\right )} e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}+\sqrt {2} \sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}\right ) c_3}{4 \sqrt {a^2 \left (a^2+4 b\right )}}+\frac {e^{-\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}-\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} \left (\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} a^2-\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} e^{\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} a^2+\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} e^{\sqrt {2} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t+\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}} a^2-\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}+\sqrt {2} \sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t} a^2-\sqrt {a^2 \left (a^2+4 b\right )} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}-\sqrt {a^2 \left (a^2+4 b\right )} \sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} e^{\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}+\sqrt {a^2 \left (a^2+4 b\right )} \sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} e^{\sqrt {2} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t+\frac {\sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}}+\sqrt {a^2 \left (a^2+4 b\right )} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} t}{\sqrt {2}}+\sqrt {2} \sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}} t}\right ) c_4}{2 \sqrt {2} \sqrt {a^2 \left (a^2+4 b\right )} \sqrt {-a^2-2 b-\sqrt {a^2 \left (a^2+4 b\right )}} \sqrt {-a^2-2 b+\sqrt {a^2 \left (a^2+4 b\right )}}}\right \}\right \}\] ✓ Maple : cpu = 0.126 (sec), leaf count = 463
\[\left \{x \relax (t ) = c_{1} {\mathrm e}^{-\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}+c_{2} {\mathrm e}^{\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}+c_{3} {\mathrm e}^{-\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}+c_{4} {\mathrm e}^{\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}, y \relax (t ) = \frac {4 c_{1} \left (\frac {\left (-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b \right )^{\frac {3}{2}}}{4}+\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, \left (a^{2}+b \right )\right ) {\mathrm e}^{-\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}-4 c_{2} \left (\frac {\left (-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b \right )^{\frac {3}{2}}}{4}+\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, \left (a^{2}+b \right )\right ) {\mathrm e}^{\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}+4 \left (c_{3} {\mathrm e}^{-\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}-c_{4} {\mathrm e}^{\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}\right ) \left (\frac {\left (-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b \right )^{\frac {3}{2}}}{4}+\left (a^{2}+b \right ) \sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\right )}{8 a b}\right \}\]