\[ \left \{x''(t)=a x(t)+b y(t),y''(t)=c x(t)+d y(t)\right \} \] ✓ Mathematica : cpu = 0.311969 (sec), leaf count = 5748
DSolve[{Derivative[2][x][t] == a*x[t] + b*y[t], Derivative[2][y][t] == c*x[t] + d*y[t]},{x[t], y[t]},t]
\[\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 = 0.12 (sec), leaf count = 360
dsolve({diff(diff(x(t),t),t) = a*x(t)+b*y(t), diff(diff(y(t),t),t) = c*x(t)+d*y(t)})
\[\left \{x \left (t \right ) = c_{1} {\mathrm e}^{-\frac {\sqrt {-2 \sqrt {a^{2}-2 d a +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+c_{2} {\mathrm e}^{\frac {\sqrt {-2 \sqrt {a^{2}-2 d a +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+c_{3} {\mathrm e}^{-\frac {\sqrt {2 \sqrt {a^{2}-2 d a +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+c_{4} {\mathrm e}^{\frac {\sqrt {2 \sqrt {a^{2}-2 d a +4 b c +d^{2}}+2 a +2 d}\, t}{2}}, y \left (t \right ) = \frac {-c_{1} \left (\sqrt {a^{2}-2 d a +4 b c +d^{2}}+a -d \right ) {\mathrm e}^{-\frac {\sqrt {-2 \sqrt {a^{2}-2 d a +4 b c +d^{2}}+2 a +2 d}\, t}{2}}-c_{2} \left (\sqrt {a^{2}-2 d a +4 b c +d^{2}}+a -d \right ) {\mathrm e}^{\frac {\sqrt {-2 \sqrt {a^{2}-2 d a +4 b c +d^{2}}+2 a +2 d}\, t}{2}}-\left (c_{3} {\mathrm e}^{-\frac {\sqrt {2 \sqrt {a^{2}-2 d a +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+c_{4} {\mathrm e}^{\frac {\sqrt {2 \sqrt {a^{2}-2 d a +4 b c +d^{2}}+2 a +2 d}\, t}{2}}\right ) \left (-\sqrt {a^{2}-2 d a +4 b c +d^{2}}+a -d \right )}{2 b}\right \}\]