\[ \left \{-a y'(t)+b x(t)+x''(t)=0,a x'(t)+b y(t)+y''(t)=0\right \} \] ✓ Mathematica : cpu = 0.250139 (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 = 0.182 (sec), leaf count = 463
\[ \left \{ \left \{ x \left ( t \right ) ={\it \_C1}\,{{\rm e}^{-{\frac {t}{2}\sqrt {-2\,{a}^{2}-2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b}}}}+{\it \_C2}\,{{\rm e}^{{\frac {t}{2}\sqrt {-2\,{a}^{2}-2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b}}}}+{\it \_C3}\,{{\rm e}^{-{\frac {t}{2}\sqrt {-2\,{a}^{2}+2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b}}}}+{\it \_C4}\,{{\rm e}^{{\frac {t}{2}\sqrt {-2\,{a}^{2}+2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b}}}},y \left ( t \right ) ={\frac {1}{8\,ab} \left ( 4\,{\it \_C1}\, \left ( 1/4\, \left ( -2\,{a}^{2}-2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b \right ) ^{3/2}+\sqrt {-2\,{a}^{2}-2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b} \left ( {a}^{2}+b \right ) \right ) {{\rm e}^{-1/2\,\sqrt {-2\,{a}^{2}-2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b}t}}-4\, \left ( 1/4\, \left ( -2\,{a}^{2}-2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b \right ) ^{3/2}+\sqrt {-2\,{a}^{2}-2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b} \left ( {a}^{2}+b \right ) \right ) {\it \_C2}\,{{\rm e}^{1/2\,\sqrt {-2\,{a}^{2}-2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b}t}}+4\, \left ( {\it \_C3}\,{{\rm e}^{-1/2\,\sqrt {-2\,{a}^{2}+2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b}t}}-{\it \_C4}\,{{\rm e}^{1/2\,\sqrt {-2\,{a}^{2}+2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b}t}} \right ) \left ( 1/4\, \left ( -2\,{a}^{2}+2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b \right ) ^{3/2}+ \left ( {a}^{2}+b \right ) \sqrt {-2\,{a}^{2}+2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b} \right ) \right ) } \right \} \right \} \]