\[ \left \{x''(t)=a x(t)+b y(t),y''(t)=c x(t)+d y(t)\right \} \] ✓ Mathematica : cpu = 0.546093 (sec), leaf count = 5647
\[\left \{\left \{x(t)\to \frac {e^{-\frac {\left (\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 ) t}{\sqrt {2}}} \left (\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}} e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} c_1+\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}} e^{\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} c_1+\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}} e^{\frac {\left (2 \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 ) t}{\sqrt {2}}} c_1+\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}} e^{\frac {\left (\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}}+2 \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}}\right ) t}{\sqrt {2}}} c_1-\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}}} 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}} e^{\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} 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}} e^{\frac {\left (2 \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 ) t}{\sqrt {2}}} 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}} e^{\frac {\left (\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}}+2 \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}}\right ) t}{\sqrt {2}}} c_2+d \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}}} \left (\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} c_1-\sqrt {2} c_2\right )+\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\left (2 \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 ) t}{\sqrt {2}}} \left (\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} c_1+\sqrt {2} c_2\right )+\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}}} \left (\sqrt {2} c_2-\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} c_1\right )-\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\left (\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}}+2 \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}}\right ) t}{\sqrt {2}}} \left (\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} c_1+\sqrt {2} c_2\right )\right )+a \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}}} \left (\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} c_1-\sqrt {2} c_2\right )+\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}}} \left (\sqrt {2} c_2-\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} c_1\right )-\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\left (2 \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 ) t}{\sqrt {2}}} \left (\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} c_1+\sqrt {2} c_2\right )+\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\left (\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}}+2 \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}}\right ) t}{\sqrt {2}}} \left (\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} c_1+\sqrt {2} c_2\right )\right )+2 b \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}}} c_3-2 b \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}}} c_3-2 b \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 {\left (2 \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 ) t}{\sqrt {2}}} c_3+2 b \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 {\left (\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}}+2 \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}}\right ) t}{\sqrt {2}}} c_3-2 \sqrt {2} b \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}}} c_4+2 \sqrt {2} b \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}}} c_4-2 \sqrt {2} b \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\left (2 \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 ) t}{\sqrt {2}}} c_4+2 \sqrt {2} b \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\left (\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}}+2 \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}}\right ) t}{\sqrt {2}}} c_4\right )}{4 \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 {e^{-\frac {\left (\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 ) t}{\sqrt {2}}} \left (2 c \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}}} \left (\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} c_1-\sqrt {2} c_2\right )+\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}}} \left (\sqrt {2} c_2-\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} c_1\right )-\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\left (2 \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 ) t}{\sqrt {2}}} \left (\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} c_1+\sqrt {2} c_2\right )+\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\left (\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}}+2 \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}}\right ) t}{\sqrt {2}}} \left (\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} c_1+\sqrt {2} c_2\right )\right )+d \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}}} c_3+\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}} e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} c_3-d \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}}} c_3+\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}} e^{\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} c_3-d \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 {\left (2 \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 ) t}{\sqrt {2}}} c_3+\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}} e^{\frac {\left (2 \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 ) t}{\sqrt {2}}} c_3+d \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 {\left (\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}}+2 \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}}\right ) t}{\sqrt {2}}} c_3+\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}} e^{\frac {\left (\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}}+2 \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}}\right ) t}{\sqrt {2}}} c_3-\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}}} 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}} e^{\frac {\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} c_4+\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}}} 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}} e^{\frac {\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} t}{\sqrt {2}}} c_4-\sqrt {2} d \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\left (2 \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 ) t}{\sqrt {2}}} 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}} e^{\frac {\left (2 \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 ) t}{\sqrt {2}}} c_4+\sqrt {2} d \sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\left (\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}}+2 \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}}\right ) t}{\sqrt {2}}} 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}} e^{\frac {\left (\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}}+2 \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}}\right ) t}{\sqrt {2}}} c_4+a \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}}} \left (\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} c_3-\sqrt {2} c_4\right )+\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\left (2 \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 ) t}{\sqrt {2}}} \left (\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} c_3+\sqrt {2} c_4\right )+\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}}} \left (\sqrt {2} c_4-\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} c_3\right )-\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}} e^{\frac {\left (\sqrt {a+d-\sqrt {a^2-2 d a+d^2+4 b c}}+2 \sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}}\right ) t}{\sqrt {2}}} \left (\sqrt {a+d+\sqrt {a^2-2 d a+d^2+4 b c}} c_3+\sqrt {2} c_4\right )\right )\right )}{4 \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 = 0.14 (sec), leaf count = 360
\[ \left \{ \left \{ x \left ( t \right ) ={\it \_C1}\,{{\rm e}^{-{\frac {t}{2}\sqrt {-2\,\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}+2\,a+2\,d}}}}+{\it \_C2}\,{{\rm e}^{{\frac {t}{2}\sqrt {-2\,\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}+2\,a+2\,d}}}}+{\it \_C3}\,{{\rm e}^{-{\frac {t}{2}\sqrt {2\,\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}+2\,a+2\,d}}}}+{\it \_C4}\,{{\rm e}^{{\frac {t}{2}\sqrt {2\,\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}+2\,a+2\,d}}}},y \left ( t \right ) ={\frac {1}{2\,b} \left ( -{\it \_C1}\, \left ( \sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}+a-d \right ) {{\rm e}^{-{\frac {t}{2}\sqrt {-2\,\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}+2\,a+2\,d}}}}-{\it \_C2}\, \left ( \sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}+a-d \right ) {{\rm e}^{{\frac {t}{2}\sqrt {-2\,\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}+2\,a+2\,d}}}}- \left ( {\it \_C3}\,{{\rm e}^{-{\frac {t}{2}\sqrt {2\,\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}+2\,a+2\,d}}}}+{\it \_C4}\,{{\rm e}^{{\frac {t}{2}\sqrt {2\,\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}+2\,a+2\,d}}}} \right ) \left ( -\sqrt {{a}^{2}-2\,da+4\,bc+{d}^{2}}+a-d \right ) \right ) } \right \} \right \} \]