\[ \boxed { \left \{ {\it a11}\,{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) +{\it b11}\,{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +{\it c11}\,x \left ( t \right ) +{\it a12}\,{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}y \left ( t \right ) +{\it b12}\,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +{\it c12}\,y \left ( t \right ) =0,{\it a21}\,{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) +{\it b21}\,{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) +{\it c21}\,x \left ( t \right ) +{\it a22}\,{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}y \left ( t \right ) +{\it b22}\,{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) +{\it c22}\,y \left ( t \right ) =0 \right \} } \]
Mathematica: cpu = 0.449557 (sec), leaf count = 7517
Result too large for latex to process
Maple: cpu = 0.156 (sec), leaf count = 1187 \[ \left \{ \left \{ x \left ( t \right ) =\sum _{{\it \_a}=1}^{4}{{\rm e}^ {{\it RootOf} \left ( \left ( {\it a22}\,{\it a11}-{\it a12}\,{\it a21} \right ) {{\it \_Z}}^{4}+ \left ( {\it a11}\,{\it b22}-{\it a12}\,{\it b21}-{\it a21}\,{\it b12}+{\it a22}\,{\it b11} \right ) {{\it \_Z}}^{3} + \left ( {\it a11}\,{\it c22}-{\it a12}\,{\it c21}-{\it a21}\,{\it c12 }+{\it a22}\,{\it c11}+{\it b11}\,{\it b22}-{\it b21}\,{\it b12} \right ) {{\it \_Z}}^{2}+ \left ( {\it b11}\,{\it c22}-{\it c21}\,{\it b12}-{\it b21}\,{\it c12}+{\it b22}\,{\it c11} \right ) {\it \_Z}+{\it c11}\,{\it c22}-{\it c12}\,{\it c21},{\it index}={\it \_a} \right ) t}} {\it \_C}_{{{\it \_a}}},y \left ( t \right ) =-{\frac { \left ( \left ( { \it a11}\,{\it a12}\,{\it a22}-{{\it a12}}^{2}{\it a21} \right ) {\it b22}+ \left ( -{\it a11}\,{{\it a22}}^{2}+{\it a12}\,{\it a21}\,{\it a22} \right ) {\it b12} \right ) \sum _{{\it \_a}=1}^{4} \left ( {\it RootOf} \left ( \left ( {\it a22}\,{\it a11}-{\it a12}\,{\it a21} \right ) {{\it \_Z}}^{4}+ \left ( {\it a11}\,{\it b22}-{\it a12}\,{\it b21}-{\it a21}\,{\it b12}+{\it a22}\,{\it b11} \right ) {{\it \_Z}}^{3} + \left ( {\it a11}\,{\it c22}-{\it a12}\,{\it c21}-{\it a21}\,{\it c12 }+{\it a22}\,{\it c11}+{\it b11}\,{\it b22}-{\it b21}\,{\it b12} \right ) {{\it \_Z}}^{2}+ \left ( {\it b11}\,{\it c22}-{\it c21}\,{\it b12}-{\it b21}\,{\it c12}+{\it b22}\,{\it c11} \right ) {\it \_Z}+{\it c11}\,{\it c22}-{\it c12}\,{\it c21},{\it index}={\it \_a} \right ) \right ) ^{3}{{\rm e}^{{\it RootOf} \left ( \left ( {\it a22}\,{\it a11 }-{\it a12}\,{\it a21} \right ) {{\it \_Z}}^{4}+ \left ( {\it a11}\,{ \it b22}-{\it a12}\,{\it b21}-{\it a21}\,{\it b12}+{\it a22}\,{\it b11 } \right ) {{\it \_Z}}^{3}+ \left ( {\it a11}\,{\it c22}-{\it a12}\,{ \it c21}-{\it a21}\,{\it c12}+{\it a22}\,{\it c11}+{\it b11}\,{\it b22 }-{\it b21}\,{\it b12} \right ) {{\it \_Z}}^{2}+ \left ( {\it b11}\,{ \it c22}-{\it c21}\,{\it b12}-{\it b21}\,{\it c12}+{\it b22}\,{\it c11 } \right ) {\it \_Z}+{\it c11}\,{\it c22}-{\it c12}\,{\it c21},{\it index}={\it \_a} \right ) t}}{\it \_C}_{{{\it \_a}}}}{{{\it a12}}^{2}{{ \it c22}}^{2}+ \left ( -2\,{\it a12}\,{\it a22}\,{\it c12}-{\it a12}\,{ \it b12}\,{\it b22}+{\it a22}\,{{\it b12}}^{2} \right ) {\it c22}+{{ \it c12}}^{2}{{\it a22}}^{2}+ \left ( {\it a12}\,{{\it b22}}^{2}-{\it a22}\,{\it b12}\,{\it b22} \right ) {\it c12}}}-{\frac { \left ( \left ( -{\it a11}\,{\it a12}\,{\it a22}+{{\it a12}}^{2}{\it a21} \right ) {\it c22}+ \left ( {\it a11}\,{{\it a22}}^{2}-{\it a12}\,{\it a21}\,{\it a22} \right ) {\it c12}+{\it a11}\,{\it a12}\,{{\it b22}}^{2 }+ \left ( -{{\it a12}}^{2}{\it b21}+ \left ( -{\it a22}\,{\it a11}-{ \it a12}\,{\it a21} \right ) {\it b12}+{\it a12}\,{\it a22}\,{\it b11} \right ) {\it b22}+{\it a12}\,{\it a22}\,{\it b12}\,{\it b21}+{\it a21 }\,{\it a22}\,{{\it b12}}^{2}-{{\it a22}}^{2}{\it b11}\,{\it b12} \right ) \sum _{{\it \_a}=1}^{4} \left ( {\it RootOf} \left ( \left ( { \it a22}\,{\it a11}-{\it a12}\,{\it a21} \right ) {{\it \_Z}}^{4}+ \left ( {\it a11}\,{\it b22}-{\it a12}\,{\it b21}-{\it a21}\,{\it b12} +{\it a22}\,{\it b11} \right ) {{\it \_Z}}^{3}+ \left ( {\it a11}\,{\it c22}-{\it a12}\,{\it c21}-{\it a21}\,{\it c12}+{\it a22}\,{\it c11}+{ \it b11}\,{\it b22}-{\it b21}\,{\it b12} \right ) {{\it \_Z}}^{2}+ \left ( {\it b11}\,{\it c22}-{\it c21}\,{\it b12}-{\it b21}\,{\it c12} +{\it b22}\,{\it c11} \right ) {\it \_Z}+{\it c11}\,{\it c22}-{\it c12} \,{\it c21},{\it index}={\it \_a} \right ) \right ) ^{2}{{\rm e}^{{\it RootOf} \left ( \left ( {\it a22}\,{\it a11}-{\it a12}\,{\it a21} \right ) {{\it \_Z}}^{4}+ \left ( {\it a11}\,{\it b22}-{\it a12}\,{\it b21}-{\it a21}\,{\it b12}+{\it a22}\,{\it b11} \right ) {{\it \_Z}}^{3} + \left ( {\it a11}\,{\it c22}-{\it a12}\,{\it c21}-{\it a21}\,{\it c12 }+{\it a22}\,{\it c11}+{\it b11}\,{\it b22}-{\it b21}\,{\it b12} \right ) {{\it \_Z}}^{2}+ \left ( {\it b11}\,{\it c22}-{\it c21}\,{\it b12}-{\it b21}\,{\it c12}+{\it b22}\,{\it c11} \right ) {\it \_Z}+{\it c11}\,{\it c22}-{\it c12}\,{\it c21},{\it index}={\it \_a} \right ) t}} {\it \_C}_{{{\it \_a}}}}{{{\it a12}}^{2}{{\it c22}}^{2}+ \left ( -2\,{ \it a12}\,{\it a22}\,{\it c12}-{\it a12}\,{\it b12}\,{\it b22}+{\it a22}\,{{\it b12}}^{2} \right ) {\it c22}+{{\it c12}}^{2}{{\it a22}}^{2} + \left ( {\it a12}\,{{\it b22}}^{2}-{\it a22}\,{\it b12}\,{\it b22} \right ) {\it c12}}}-{\frac { \left ( \left ( {{\it a12}}^{2}{\it b21}- {\it a12}\,{\it a22}\,{\it b11} \right ) {\it c22}+ \left ( -{{\it a12}} ^{2}{\it b22}+{\it a12}\,{\it a22}\,{\it b12} \right ) {\it c21}+ \left ( -{\it a12}\,{\it a22}\,{\it b21}+{{\it a22}}^{2}{\it b11} \right ) {\it c12}+ \left ( {\it a12}\,{\it a22}\,{\it b22}-{{\it a22}} ^{2}{\it b12} \right ) {\it c11}+{\it a12}\,{\it b11}\,{{\it b22}}^{2}+ \left ( -{\it a12}\,{\it b12}\,{\it b21}-{\it a22}\,{\it b11}\,{\it b12} \right ) {\it b22}+{\it a22}\,{\it b21}\,{{\it b12}}^{2} \right ) \sum _{{\it \_a}=1}^{4}{\it RootOf} \left ( \left ( {\it a22}\,{\it a11 }-{\it a12}\,{\it a21} \right ) {{\it \_Z}}^{4}+ \left ( {\it a11}\,{ \it b22}-{\it a12}\,{\it b21}-{\it a21}\,{\it b12}+{\it a22}\,{\it b11 } \right ) {{\it \_Z}}^{3}+ \left ( {\it a11}\,{\it c22}-{\it a12}\,{ \it c21}-{\it a21}\,{\it c12}+{\it a22}\,{\it c11}+{\it b11}\,{\it b22 }-{\it b21}\,{\it b12} \right ) {{\it \_Z}}^{2}+ \left ( {\it b11}\,{ \it c22}-{\it c21}\,{\it b12}-{\it b21}\,{\it c12}+{\it b22}\,{\it c11 } \right ) {\it \_Z}+{\it c11}\,{\it c22}-{\it c12}\,{\it c21},{\it index}={\it \_a} \right ) {{\rm e}^{{\it RootOf} \left ( \left ( {\it a22}\,{\it a11}-{\it a12}\,{\it a21} \right ) {{\it \_Z}}^{4}+ \left ( { \it a11}\,{\it b22}-{\it a12}\,{\it b21}-{\it a21}\,{\it b12}+{\it a22 }\,{\it b11} \right ) {{\it \_Z}}^{3}+ \left ( {\it a11}\,{\it c22}-{ \it a12}\,{\it c21}-{\it a21}\,{\it c12}+{\it a22}\,{\it c11}+{\it b11 }\,{\it b22}-{\it b21}\,{\it b12} \right ) {{\it \_Z}}^{2}+ \left ( { \it b11}\,{\it c22}-{\it c21}\,{\it b12}-{\it b21}\,{\it c12}+{\it b22 }\,{\it c11} \right ) {\it \_Z}+{\it c11}\,{\it c22}-{\it c12}\,{\it c21},{\it index}={\it \_a} \right ) t}}{\it \_C}_{{{\it \_a}}}}{{{\it a12}}^{2}{{\it c22}}^{2}+ \left ( -2\,{\it a12}\,{\it a22}\,{\it c12}-{ \it a12}\,{\it b12}\,{\it b22}+{\it a22}\,{{\it b12}}^{2} \right ) { \it c22}+{{\it c12}}^{2}{{\it a22}}^{2}+ \left ( {\it a12}\,{{\it b22}} ^{2}-{\it a22}\,{\it b12}\,{\it b22} \right ) {\it c12}}}-{\frac { \left ( \left ( {{\it a12}}^{2}{\it c21}-{\it a12}\,{\it a22}\,{\it c11} \right ) {\it c22}+ \left ( -{\it a12}\,{\it a22}\,{\it c12}-{\it a12}\,{\it b12}\,{\it b22}+{\it a22}\,{{\it b12}}^{2} \right ) {\it c21 }+{{\it a22}}^{2}{\it c11}\,{\it c12}+ \left ( {\it a12}\,{{\it b22}}^{ 2}-{\it a22}\,{\it b12}\,{\it b22} \right ) {\it c11} \right ) \sum _{{ \it \_a}=1}^{4}{{\rm e}^{{\it RootOf} \left ( \left ( {\it a22}\,{\it a11}-{\it a12}\,{\it a21} \right ) {{\it \_Z}}^{4}+ \left ( {\it a11}\,{ \it b22}-{\it a12}\,{\it b21}-{\it a21}\,{\it b12}+{\it a22}\,{\it b11 } \right ) {{\it \_Z}}^{3}+ \left ( {\it a11}\,{\it c22}-{\it a12}\,{ \it c21}-{\it a21}\,{\it c12}+{\it a22}\,{\it c11}+{\it b11}\,{\it b22 }-{\it b21}\,{\it b12} \right ) {{\it \_Z}}^{2}+ \left ( {\it b11}\,{ \it c22}-{\it c21}\,{\it b12}-{\it b21}\,{\it c12}+{\it b22}\,{\it c11 } \right ) {\it \_Z}+{\it c11}\,{\it c22}-{\it c12}\,{\it c21},{\it index}={\it \_a} \right ) t}}{\it \_C}_{{{\it \_a}}}}{{{\it a12}}^{2}{{ \it c22}}^{2}+ \left ( -2\,{\it a12}\,{\it a22}\,{\it c12}-{\it a12}\,{ \it b12}\,{\it b22}+{\it a22}\,{{\it b12}}^{2} \right ) {\it c22}+{{ \it c12}}^{2}{{\it a22}}^{2}+ \left ( {\it a12}\,{{\it b22}}^{2}-{\it a22}\,{\it b12}\,{\it b22} \right ) {\it c12}}} \right \} \right \} \]