\[ \left \{x''(t)-x'(t)+y'(t)=0,x''(t)-x(t)+y''(t)=0\right \} \] ✓ Mathematica : cpu = 0.0252624 (sec), leaf count = 420
DSolve[{-Derivative[1][x][t] + Derivative[1][y][t] + Derivative[2][x][t] == 0, -x[t] + Derivative[2][x][t] + Derivative[2][y][t] == 0},{x[t], y[t]},t]
\[\left \{\left \{x(t)\to -\frac {1}{5} c_1 e^{\frac {t}{2}-\frac {\sqrt {5} t}{2}} \left (\sqrt {5} e^{\sqrt {5} t}-5 e^{\frac {\sqrt {5} t}{2}+\frac {t}{2}}-\sqrt {5}\right )+\frac {c_2 e^{\frac {t}{2}-\frac {\sqrt {5} t}{2}} \left (e^{\sqrt {5} t}-1\right )}{\sqrt {5}}-\frac {1}{10} c_4 e^{\frac {t}{2}-\frac {\sqrt {5} t}{2}} \left (5 e^{\sqrt {5} t}+\sqrt {5} e^{\sqrt {5} t}-10 e^{\frac {\sqrt {5} t}{2}+\frac {t}{2}}+5-\sqrt {5}\right ),y(t)\to -\frac {1}{10} c_1 e^{-\frac {\sqrt {5} t}{2}} \left (-5 e^{t/2}-\sqrt {5} e^{t/2}+10 e^{\frac {\sqrt {5} t}{2}}-5 e^{\sqrt {5} t+\frac {t}{2}}+\sqrt {5} e^{\sqrt {5} t+\frac {t}{2}}\right )+\frac {1}{10} c_2 e^{-\frac {\sqrt {5} t}{2}} \left (-5 e^{t/2}-\sqrt {5} e^{t/2}+10 e^{\frac {\sqrt {5} t}{2}}-5 e^{\sqrt {5} t+\frac {t}{2}}+\sqrt {5} e^{\sqrt {5} t+\frac {t}{2}}\right )+\frac {c_4 e^{\frac {t}{2}-\frac {\sqrt {5} t}{2}} \left (e^{\sqrt {5} t}-1\right )}{\sqrt {5}}+c_3\right \}\right \}\] ✓ Maple : cpu = 0.079 (sec), leaf count = 71
dsolve({diff(diff(x(t),t),t)+diff(diff(y(t),t),t)-x(t) = 0, diff(diff(x(t),t),t)-diff(x(t),t)+diff(y(t),t) = 0})
\[\left \{x \left (t \right ) = \frac {c_{4} \left (\sqrt {5}-1\right ) {\mathrm e}^{-\frac {\left (\sqrt {5}-1\right ) t}{2}}}{2}-\frac {c_{3} \left (\sqrt {5}+1\right ) {\mathrm e}^{\frac {\left (\sqrt {5}+1\right ) t}{2}}}{2}+c_{1} {\mathrm e}^{t}, y \left (t \right ) = c_{2}+c_{3} {\mathrm e}^{\frac {\left (\sqrt {5}+1\right ) t}{2}}+c_{4} {\mathrm e}^{-\frac {\left (\sqrt {5}-1\right ) t}{2}}\right \}\]