\[ \left \{-a y'(t)+b x(t)+x''(t)=0,a x'(t)+b y(t)+y''(t)=0\right \} \] ✓ Mathematica : cpu = 0.408204 (sec), leaf count = 3522
\[\left \{\left \{x(t)\to \frac {e^{-\frac {\left (\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}+\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}}\right ) t}{\sqrt {2}}} \left (\left (\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} e^{\frac {\left (\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}+2 \sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}}\right ) t}{\sqrt {2}}} \left (\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} c_1-\sqrt {2} c_2\right )+\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} e^{\frac {\left (2 \sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}+\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}}\right ) t}{\sqrt {2}}} \left (\sqrt {2} c_2-\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} c_1\right )-\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} e^{\frac {\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} t}{\sqrt {2}}} \left (\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} c_1+\sqrt {2} c_2\right )+\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} t}{\sqrt {2}}} \left (\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} c_1+\sqrt {2} c_2\right )\right ) a^2+\left (2 \sqrt {2} b \left (e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} t}{\sqrt {2}}} \sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}-e^{\frac {\left (\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}+2 \sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}}\right ) t}{\sqrt {2}}} \sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}-\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} e^{\frac {\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} t}{\sqrt {2}}}+\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} e^{\frac {\left (2 \sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}+\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}}\right ) t}{\sqrt {2}}}\right ) c_3+2 \sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} \sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} \left (e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} t}{\sqrt {2}}}-e^{\frac {\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} t}{\sqrt {2}}}-e^{\frac {\left (2 \sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}+\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}}\right ) t}{\sqrt {2}}}+e^{\frac {\left (\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}+2 \sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}}\right ) t}{\sqrt {2}}}\right ) c_4\right ) a+\sqrt {a^4+4 b a^2} \left (\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} e^{\frac {\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} t}{\sqrt {2}}} \left (\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} c_1-\sqrt {2} c_2\right )+\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} t}{\sqrt {2}}} \left (\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} c_1-\sqrt {2} c_2\right )+\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} e^{\frac {\left (2 \sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}+\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}}\right ) t}{\sqrt {2}}} \left (\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} c_1+\sqrt {2} c_2\right )+\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} e^{\frac {\left (\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}+2 \sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}}\right ) t}{\sqrt {2}}} \left (\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} c_1+\sqrt {2} c_2\right )\right )\right )}{4 \sqrt {a^4+4 b a^2} \sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} \sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}}},y(t)\to \frac {e^{-\frac {\left (\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}+\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}}\right ) t}{\sqrt {2}}} \left (\left (\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} e^{\frac {\left (\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}+2 \sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}}\right ) t}{\sqrt {2}}} \left (\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} c_3-\sqrt {2} c_4\right )+\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} e^{\frac {\left (2 \sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}+\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}}\right ) t}{\sqrt {2}}} \left (\sqrt {2} c_4-\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} c_3\right )-\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} e^{\frac {\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} t}{\sqrt {2}}} \left (\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} c_3+\sqrt {2} c_4\right )+\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} t}{\sqrt {2}}} \left (\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} c_3+\sqrt {2} c_4\right )\right ) a^2+\left (-2 \sqrt {2} b \left (e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} t}{\sqrt {2}}} \sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}-e^{\frac {\left (\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}+2 \sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}}\right ) t}{\sqrt {2}}} \sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}-\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} e^{\frac {\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} t}{\sqrt {2}}}+\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} e^{\frac {\left (2 \sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}+\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}}\right ) t}{\sqrt {2}}}\right ) c_1-2 \sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} \sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} \left (e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} t}{\sqrt {2}}}-e^{\frac {\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} t}{\sqrt {2}}}-e^{\frac {\left (2 \sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}+\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}}\right ) t}{\sqrt {2}}}+e^{\frac {\left (\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}+2 \sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}}\right ) t}{\sqrt {2}}}\right ) c_2\right ) a+\sqrt {a^4+4 b a^2} \left (\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} e^{\frac {\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} t}{\sqrt {2}}} \left (\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} c_3-\sqrt {2} c_4\right )+\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} e^{\frac {\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} t}{\sqrt {2}}} \left (\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} c_3-\sqrt {2} c_4\right )+\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} e^{\frac {\left (2 \sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}+\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}}\right ) t}{\sqrt {2}}} \left (\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} c_3+\sqrt {2} c_4\right )+\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} e^{\frac {\left (\sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}}+2 \sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}}\right ) t}{\sqrt {2}}} \left (\sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}} c_3+\sqrt {2} c_4\right )\right )\right )}{4 \sqrt {a^4+4 b a^2} \sqrt {-a^2-2 b-\sqrt {a^4+4 b a^2}} \sqrt {-a^2-2 b+\sqrt {a^4+4 b a^2}}}\right \}\right \}\]
✓ Maple : cpu = 0.151 (sec), leaf count = 463
\[ \left \{ \left \{ x \left ( t \right ) ={\it \_C1}\,{{\rm e}^{-{\frac {t}{2}\sqrt {-2\,{a}^{2}-2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b}}}}+{\it \_C2}\,{{\rm e}^{{\frac {t}{2}\sqrt {-2\,{a}^{2}-2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b}}}}+{\it \_C3}\,{{\rm e}^{-{\frac {t}{2}\sqrt {-2\,{a}^{2}+2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b}}}}+{\it \_C4}\,{{\rm e}^{{\frac {t}{2}\sqrt {-2\,{a}^{2}+2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b}}}},y \left ( t \right ) ={\frac {1}{8\,ab} \left ( 4\, \left ( 1/4\, \left ( -2\,{a}^{2}-2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b \right ) ^{3/2}+\sqrt {-2\,{a}^{2}-2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b} \left ( {a}^{2}+b \right ) \right ) {\it \_C1}\,{{\rm e}^{-1/2\,\sqrt {-2\,{a}^{2}-2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b}t}}-4\, \left ( 1/4\, \left ( -2\,{a}^{2}-2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b \right ) ^{3/2}+\sqrt {-2\,{a}^{2}-2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b} \left ( {a}^{2}+b \right ) \right ) {\it \_C2}\,{{\rm e}^{1/2\,\sqrt {-2\,{a}^{2}-2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b}t}}+4\, \left ( {\it \_C3}\,{{\rm e}^{-1/2\,\sqrt {-2\,{a}^{2}+2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b}t}}-{\it \_C4}\,{{\rm e}^{1/2\,\sqrt {-2\,{a}^{2}+2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b}t}} \right ) \left ( 1/4\, \left ( -2\,{a}^{2}+2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b \right ) ^{3/2}+ \left ( {a}^{2}+b \right ) \sqrt {-2\,{a}^{2}+2\,\sqrt {{a}^{2} \left ( {a}^{2}+4\,b \right ) }-4\,b} \right ) \right ) } \right \} \right \} \]