\[ \left \{x''(t)=a x(t)+b y(t),y''(t)=c x(t)+d y(t)\right \} \] ✓ Mathematica : cpu = 0.532388 (sec), leaf count = 5748
\[\left \{\left \{x(t)\to \frac {e^{-\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}-\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} \left (e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} a-e^{\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} a-e^{\sqrt {2} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t+\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} a+e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}+\sqrt {2} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t} a-d e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}+\sqrt {a^2-2 d a+d^2+4 b c} e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}+d e^{\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}+\sqrt {a^2-2 d a+d^2+4 b c} e^{\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}+d e^{\sqrt {2} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t+\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}+\sqrt {a^2-2 d a+d^2+4 b c} e^{\sqrt {2} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t+\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}-d e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}+\sqrt {2} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}+\sqrt {a^2-2 d a+d^2+4 b c} e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}+\sqrt {2} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}\right ) c_1}{4 \sqrt {a^2-2 d a+d^2+4 b c}}+\frac {e^{-\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}-\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} \left (-\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} a+\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} a-\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\sqrt {2} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t+\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} a+\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}+\sqrt {2} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t} a+d \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}-\sqrt {a^2-2 d a+d^2+4 b c} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}-d \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}-\sqrt {a^2-2 d a+d^2+4 b c} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}+d \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\sqrt {2} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t+\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}+\sqrt {a^2-2 d a+d^2+4 b c} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\sqrt {2} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t+\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}-d \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}+\sqrt {2} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}+\sqrt {a^2-2 d a+d^2+4 b c} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}+\sqrt {2} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}\right ) c_2}{2 \sqrt {2} \sqrt {a^2-2 d a+d^2+4 b c} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}}}-\frac {b e^{-\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}-\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} \left (-e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}+e^{\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}+e^{\sqrt {2} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t+\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}-e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}+\sqrt {2} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}\right ) c_3}{2 \sqrt {a^2-2 d a+d^2+4 b c}}+\frac {b e^{-\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}-\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} \left (-e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}}+e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}+\sqrt {2} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}}+\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}-\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\sqrt {2} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t+\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}\right ) c_4}{\sqrt {2} \sqrt {a^2-2 d a+d^2+4 b c} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}}},y(t)\to -\frac {c e^{-\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}-\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} \left (-e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}+e^{\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}+e^{\sqrt {2} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t+\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}-e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}+\sqrt {2} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}\right ) c_1}{2 \sqrt {a^2-2 d a+d^2+4 b c}}+\frac {c e^{-\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}-\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} \left (-e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}}+e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}+\sqrt {2} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}}+\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}-\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\sqrt {2} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t+\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}\right ) c_2}{\sqrt {2} \sqrt {a^2-2 d a+d^2+4 b c} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}}}+\frac {e^{-\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}-\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} \left (-e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} a+e^{\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} a+e^{\sqrt {2} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t+\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} a-e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}+\sqrt {2} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t} a+d e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}+\sqrt {a^2-2 d a+d^2+4 b c} e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}-d e^{\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}+\sqrt {a^2-2 d a+d^2+4 b c} e^{\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}-d e^{\sqrt {2} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t+\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}+\sqrt {a^2-2 d a+d^2+4 b c} e^{\sqrt {2} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t+\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}+d e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}+\sqrt {2} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}+\sqrt {a^2-2 d a+d^2+4 b c} e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}+\sqrt {2} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}\right ) c_3}{4 \sqrt {a^2-2 d a+d^2+4 b c}}+\frac {e^{-\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}-\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} \left (\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} a-\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} a+\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\sqrt {2} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t+\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} a-\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}+\sqrt {2} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t} a-d \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}-\sqrt {a^2-2 d a+d^2+4 b c} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}+d \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}-\sqrt {a^2-2 d a+d^2+4 b c} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}-d \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\sqrt {2} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t+\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}+\sqrt {a^2-2 d a+d^2+4 b c} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\sqrt {2} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t+\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}}+d \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}+\sqrt {2} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}+\sqrt {a^2-2 d a+d^2+4 b c} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}+\sqrt {2} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}\right ) c_4}{2 \sqrt {2} \sqrt {a^2-2 d a+d^2+4 b c} \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}}}\right \}\right \}\]
✓ Maple : cpu = 0.127 (sec), leaf count = 418
\[ \left \{ \left \{ x \left ( t \right ) ={\it \_C1}\,{{\rm e}^{-{\frac {t}{2}\sqrt {-2\,\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}+2\,a+2\,d}}}}+{\it \_C2}\,{{\rm e}^{{\frac {t}{2}\sqrt {-2\,\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}+2\,a+2\,d}}}}+{\it \_C3}\,{{\rm e}^{-{\frac {t}{2}\sqrt {2\,\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}+2\,a+2\,d}}}}+{\it \_C4}\,{{\rm e}^{{\frac {t}{2}\sqrt {2\,\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}+2\,a+2\,d}}}},y \left ( t \right ) = \left ( {\frac {d}{2\,b}}+{\frac {1}{b} \left ( {\frac {1}{2}\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}}-{\frac {a}{2}} \right ) } \right ) {\it \_C4}\,{{\rm e}^{{\frac {t}{2}\sqrt {2\,\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}+2\,a+2\,d}}}}+ \left ( {\frac {d}{2\,b}}+{\frac {1}{b} \left ( {\frac {1}{2}\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}}-{\frac {a}{2}} \right ) } \right ) {\it \_C3}\,{{\rm e}^{-{\frac {t}{2}\sqrt {2\,\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}+2\,a+2\,d}}}}+ \left ( {\frac {d}{2\,b}}+{\frac {1}{b} \left ( -{\frac {1}{2}\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}}-{\frac {a}{2}} \right ) } \right ) {\it \_C2}\,{{\rm e}^{{\frac {t}{2}\sqrt {-2\,\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}+2\,a+2\,d}}}}+ \left ( {\frac {d}{2\,b}}+{\frac {1}{b} \left ( -{\frac {1}{2}\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}}-{\frac {a}{2}} \right ) } \right ) {\it \_C1}\,{{\rm e}^{-{\frac {t}{2}\sqrt {-2\,\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}+2\,a+2\,d}}}} \right \} \right \} \]