\[ \boxed { \left \{ {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) +a \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) -{\frac {\rm d}{{\rm d}t}}y \left ( t \right ) \right ) +{\it b1}\,x \left ( t \right ) ={\it c1}\,{{\rm e}^{i\omega \,t}},{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}y \left ( t \right ) +a \left ( {\frac {\rm d}{{\rm d}t}}y \left ( t \right ) -{\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) +{\it b2}\,y \left ( t \right ) ={\it c2}\,{{\rm e}^{i\omega \,t}} \right \} } \]
Mathematica: cpu = 0 (sec), leaf count = 0 \[ \text {Hanged} \]
Maple: cpu = 0.671 (sec), leaf count = 2511 \[ \left \{ \left \{ x \left ( t \right ) ={\frac {i{{\rm e}^{i\omega \,t}}a {\it c1}\,\omega +i{{\rm e}^{i\omega \,t}}a{\it c2}\,\omega -{{\rm e}^{i \omega \,t}}{\it c1}\,{\omega }^{2}+{{\rm e}^{i\omega \,t}}{\it b2}\,{ \it c1}}{-2\,ia{\omega }^{3}+ia{\it b1}\,\omega +ia{\it b2}\,\omega +{ \omega }^{4}-{\it b1}\,{\omega }^{2}-{\it b2}\,{\omega }^{2}+{\it b2}\,{ \it b1}}}+{\it \_C1}\,{{\rm e}^{{\it RootOf} \left ( {{\it \_Z}}^{4}+2 \,a{{\it \_Z}}^{3}+ \left ( {\it b1}+{\it b2} \right ) {{\it \_Z}}^{2}+ \left ( a{\it b1}+{\it b2}\,a \right ) {\it \_Z}+{\it b2}\,{\it b1},{ \it index}=1 \right ) t}}+{\it \_C2}\,{{\rm e}^{{\it RootOf} \left ( {{ \it \_Z}}^{4}+2\,a{{\it \_Z}}^{3}+ \left ( {\it b1}+{\it b2} \right ) {{ \it \_Z}}^{2}+ \left ( a{\it b1}+{\it b2}\,a \right ) {\it \_Z}+{\it b2} \,{\it b1},{\it index}=2 \right ) t}}+{\it \_C3}\,{{\rm e}^{{\it RootOf } \left ( {{\it \_Z}}^{4}+2\,a{{\it \_Z}}^{3}+ \left ( {\it b1}+{\it b2} \right ) {{\it \_Z}}^{2}+ \left ( a{\it b1}+{\it b2}\,a \right ) {\it \_Z}+{\it b2}\,{\it b1},{\it index}=3 \right ) t}}+{\it \_C4}\,{{\rm e} ^{{\it RootOf} \left ( {{\it \_Z}}^{4}+2\,a{{\it \_Z}}^{3}+ \left ( { \it b1}+{\it b2} \right ) {{\it \_Z}}^{2}+ \left ( a{\it b1}+{\it b2}\,a \right ) {\it \_Z}+{\it b2}\,{\it b1},{\it index}=4 \right ) t}},y \left ( t \right ) = \left ( {\frac { \left ( -{\it \_C4}\,{\omega }^{4}+2 \,i{\it \_C4}\,a{\omega }^{3}+ \left ( {\it b1}+{\it b2} \right ) {\it \_C4}\,{\omega }^{2}+ \left ( -i{\it b1}\,a-i{\it b2}\,a \right ) {\it \_C4}\,\omega -{\it \_C4}\,{\it b2}\,{\it b1} \right ) \left ( {\it RootOf} \left ( {{\it \_Z}}^{4}+2\,a{{\it \_Z}}^{3}+ \left ( {\it b1}+{ \it b2} \right ) {{\it \_Z}}^{2}+ \left ( a{\it b1}+{\it b2}\,a \right ) {\it \_Z}+{\it b2}\,{\it b1},{\it index}=4 \right ) \right ) ^{3}}{ \left ( {\omega }^{4}-2\,ia{\omega }^{3}+ \left ( -{\it b1}-{\it b2} \right ) {\omega }^{2}+ \left ( i{\it b1}\,a+i{\it b2}\,a \right ) \omega +{\it b2}\,{\it b1} \right ) {\it b2}\,a}}+{\frac { \left ( -2\,{\it \_C4}\,a{\omega }^{4}+4\,i{\it \_C4}\,{a}^{2}{\omega }^{3}+ \left ( 2\,a{ \it b1}+2\,{\it b2}\,a \right ) {\it \_C4}\,{\omega }^{2}+ \left ( -2\,i{ a}^{2}{\it b1}-2\,i{a}^{2}{\it b2} \right ) {\it \_C4}\,\omega -2\,{\it \_C4}\,a{\it b1}\,{\it b2} \right ) \left ( {\it RootOf} \left ( {{\it \_Z}}^{4}+2\,a{{\it \_Z}}^{3}+ \left ( {\it b1}+{\it b2} \right ) {{\it \_Z}}^{2}+ \left ( a{\it b1}+{\it b2}\,a \right ) {\it \_Z}+{\it b2}\,{ \it b1},{\it index}=4 \right ) \right ) ^{2}}{ \left ( {\omega }^{4}-2\,i a{\omega }^{3}+ \left ( -{\it b1}-{\it b2} \right ) {\omega }^{2}+ \left ( i{\it b1}\,a+i{\it b2}\,a \right ) \omega +{\it b2}\,{\it b1} \right ) { \it b2}\,a}}+{\frac { \left ( -{\it \_C4}\,{\it b1}\,{\omega }^{4}+2\,i{ \it \_C4}\,a{\it b1}\,{\omega }^{3}+ \left ( {{\it b1}}^{2}+{\it b2}\,{ \it b1} \right ) {\it \_C4}\,{\omega }^{2}+ \left ( -ia{{\it b1}}^{2}-ia{ \it b1}\,{\it b2} \right ) {\it \_C4}\,\omega -{\it \_C4}\,{{\it b1}}^{2 }{\it b2} \right ) {\it RootOf} \left ( {{\it \_Z}}^{4}+2\,a{{\it \_Z}}^ {3}+ \left ( {\it b1}+{\it b2} \right ) {{\it \_Z}}^{2}+ \left ( a{\it b1 }+{\it b2}\,a \right ) {\it \_Z}+{\it b2}\,{\it b1},{\it index}=4 \right ) }{ \left ( {\omega }^{4}-2\,ia{\omega }^{3}+ \left ( -{\it b1}-{ \it b2} \right ) {\omega }^{2}+ \left ( i{\it b1}\,a+i{\it b2}\,a \right ) \omega +{\it b2}\,{\it b1} \right ) {\it b2}\,a}}+{\frac {-{ \it \_C4}\,a{\it b1}\,{\omega }^{4}+2\,i{\it \_C4}\,{a}^{2}{\it b1}\,{ \omega }^{3}+ \left ( a{{\it b1}}^{2}+a{\it b1}\,{\it b2} \right ) {\it \_C4}\,{\omega }^{2}+ \left ( -i{a}^{2}{{\it b1}}^{2}-i{a}^{2}{\it b1}\, {\it b2} \right ) {\it \_C4}\,\omega -{\it \_C4}\,a{{\it b1}}^{2}{\it b2 }}{ \left ( {\omega }^{4}-2\,ia{\omega }^{3}+ \left ( -{\it b1}-{\it b2} \right ) {\omega }^{2}+ \left ( i{\it b1}\,a+i{\it b2}\,a \right ) \omega +{\it b2}\,{\it b1} \right ) {\it b2}\,a}} \right ) {{\rm e}^{{\it RootOf} \left ( {{\it \_Z}}^{4}+2\,a{{\it \_Z}}^{3}+ \left ( {\it b1}+{ \it b2} \right ) {{\it \_Z}}^{2}+ \left ( a{\it b1}+{\it b2}\,a \right ) {\it \_Z}+{\it b2}\,{\it b1},{\it index}=4 \right ) t}}+ \left ( {\frac { \left ( -{\it \_C3}\,{\omega }^{4}+2\,i{\it \_C3}\,a{\omega }^{3}+ \left ( {\it b1}+{\it b2} \right ) {\it \_C3}\,{\omega }^{2}+ \left ( -i{ \it b1}\,a-i{\it b2}\,a \right ) {\it \_C3}\,\omega -{\it \_C3}\,{\it b2 }\,{\it b1} \right ) \left ( {\it RootOf} \left ( {{\it \_Z}}^{4}+2\,a{{ \it \_Z}}^{3}+ \left ( {\it b1}+{\it b2} \right ) {{\it \_Z}}^{2}+ \left ( a{\it b1}+{\it b2}\,a \right ) {\it \_Z}+{\it b2}\,{\it b1},{ \it index}=3 \right ) \right ) ^{3}}{ \left ( {\omega }^{4}-2\,ia{\omega } ^{3}+ \left ( -{\it b1}-{\it b2} \right ) {\omega }^{2}+ \left ( i{\it b1} \,a+i{\it b2}\,a \right ) \omega +{\it b2}\,{\it b1} \right ) {\it b2}\,a }}+{\frac { \left ( -2\,{\it \_C3}\,a{\omega }^{4}+4\,i{\it \_C3}\,{a}^{ 2}{\omega }^{3}+ \left ( 2\,a{\it b1}+2\,{\it b2}\,a \right ) {\it \_C3} \,{\omega }^{2}+ \left ( -2\,i{a}^{2}{\it b1}-2\,i{a}^{2}{\it b2} \right ) {\it \_C3}\,\omega -2\,{\it \_C3}\,a{\it b1}\,{\it b2} \right ) \left ( {\it RootOf} \left ( {{\it \_Z}}^{4}+2\,a{{\it \_Z}}^{ 3}+ \left ( {\it b1}+{\it b2} \right ) {{\it \_Z}}^{2}+ \left ( a{\it b1} +{\it b2}\,a \right ) {\it \_Z}+{\it b2}\,{\it b1},{\it index}=3 \right ) \right ) ^{2}}{ \left ( {\omega }^{4}-2\,ia{\omega }^{3}+ \left ( -{\it b1}-{\it b2} \right ) {\omega }^{2}+ \left ( i{\it b1}\,a+i {\it b2}\,a \right ) \omega +{\it b2}\,{\it b1} \right ) {\it b2}\,a}}+{ \frac { \left ( -{\it \_C3}\,{\it b1}\,{\omega }^{4}+2\,i{\it \_C3}\,a{ \it b1}\,{\omega }^{3}+ \left ( {{\it b1}}^{2}+{\it b2}\,{\it b1} \right ) {\it \_C3}\,{\omega }^{2}+ \left ( -ia{{\it b1}}^{2}-ia{\it b1} \,{\it b2} \right ) {\it \_C3}\,\omega -{\it \_C3}\,{{\it b1}}^{2}{\it b2} \right ) {\it RootOf} \left ( {{\it \_Z}}^{4}+2\,a{{\it \_Z}}^{3}+ \left ( {\it b1}+{\it b2} \right ) {{\it \_Z}}^{2}+ \left ( a{\it b1}+{ \it b2}\,a \right ) {\it \_Z}+{\it b2}\,{\it b1},{\it index}=3 \right ) }{ \left ( {\omega }^{4}-2\,ia{\omega }^{3}+ \left ( -{\it b1}-{\it b2} \right ) {\omega }^{2}+ \left ( i{\it b1}\,a+i{\it b2}\,a \right ) \omega +{\it b2}\,{\it b1} \right ) {\it b2}\,a}}+{\frac {-{\it \_C3}\,a{\it b1}\,{\omega }^{4}+2\,i{\it \_C3}\,{a}^{2}{\it b1}\,{\omega }^{3}+ \left ( a{{\it b1}}^{2}+a{\it b1}\,{\it b2} \right ) {\it \_C3}\,{ \omega }^{2}+ \left ( -i{a}^{2}{{\it b1}}^{2}-i{a}^{2}{\it b1}\,{\it b2} \right ) {\it \_C3}\,\omega -{\it \_C3}\,a{{\it b1}}^{2}{\it b2}}{ \left ( {\omega }^{4}-2\,ia{\omega }^{3}+ \left ( -{\it b1}-{\it b2} \right ) {\omega }^{2}+ \left ( i{\it b1}\,a+i{\it b2}\,a \right ) \omega +{\it b2}\,{\it b1} \right ) {\it b2}\,a}} \right ) {{\rm e}^{{\it RootOf} \left ( {{\it \_Z}}^{4}+2\,a{{\it \_Z}}^{3}+ \left ( {\it b1}+{ \it b2} \right ) {{\it \_Z}}^{2}+ \left ( a{\it b1}+{\it b2}\,a \right ) {\it \_Z}+{\it b2}\,{\it b1},{\it index}=3 \right ) t}}+ \left ( {\frac { \left ( -{\it \_C2}\,{\omega }^{4}+2\,i{\it \_C2}\,a{\omega }^{3}+ \left ( {\it b1}+{\it b2} \right ) {\it \_C2}\,{\omega }^{2}+ \left ( -i{ \it b1}\,a-i{\it b2}\,a \right ) {\it \_C2}\,\omega -{\it \_C2}\,{\it b1 }\,{\it b2} \right ) \left ( {\it RootOf} \left ( {{\it \_Z}}^{4}+2\,a{{ \it \_Z}}^{3}+ \left ( {\it b1}+{\it b2} \right ) {{\it \_Z}}^{2}+ \left ( a{\it b1}+{\it b2}\,a \right ) {\it \_Z}+{\it b2}\,{\it b1},{ \it index}=2 \right ) \right ) ^{3}}{ \left ( {\omega }^{4}-2\,ia{\omega } ^{3}+ \left ( -{\it b1}-{\it b2} \right ) {\omega }^{2}+ \left ( i{\it b1} \,a+i{\it b2}\,a \right ) \omega +{\it b2}\,{\it b1} \right ) {\it b2}\,a }}+{\frac { \left ( -2\,{\it \_C2}\,a{\omega }^{4}+4\,i{\it \_C2}\,{a}^{ 2}{\omega }^{3}+ \left ( 2\,a{\it b1}+2\,{\it b2}\,a \right ) {\it \_C2} \,{\omega }^{2}+ \left ( -2\,i{a}^{2}{\it b1}-2\,i{a}^{2}{\it b2} \right ) {\it \_C2}\,\omega -2\,{\it \_C2}\,a{\it b1}\,{\it b2} \right ) \left ( {\it RootOf} \left ( {{\it \_Z}}^{4}+2\,a{{\it \_Z}}^{ 3}+ \left ( {\it b1}+{\it b2} \right ) {{\it \_Z}}^{2}+ \left ( a{\it b1} +{\it b2}\,a \right ) {\it \_Z}+{\it b2}\,{\it b1},{\it index}=2 \right ) \right ) ^{2}}{ \left ( {\omega }^{4}-2\,ia{\omega }^{3}+ \left ( -{\it b1}-{\it b2} \right ) {\omega }^{2}+ \left ( i{\it b1}\,a+i {\it b2}\,a \right ) \omega +{\it b2}\,{\it b1} \right ) {\it b2}\,a}}+{ \frac { \left ( -{\it \_C2}\,{\it b1}\,{\omega }^{4}+2\,i{\it \_C2}\,a{ \it b1}\,{\omega }^{3}+ \left ( {{\it b1}}^{2}+{\it b2}\,{\it b1} \right ) {\it \_C2}\,{\omega }^{2}+ \left ( -ia{{\it b1}}^{2}-ia{\it b1} \,{\it b2} \right ) {\it \_C2}\,\omega -{\it \_C2}\,{{\it b1}}^{2}{\it b2} \right ) {\it RootOf} \left ( {{\it \_Z}}^{4}+2\,a{{\it \_Z}}^{3}+ \left ( {\it b1}+{\it b2} \right ) {{\it \_Z}}^{2}+ \left ( a{\it b1}+{ \it b2}\,a \right ) {\it \_Z}+{\it b2}\,{\it b1},{\it index}=2 \right ) }{ \left ( {\omega }^{4}-2\,ia{\omega }^{3}+ \left ( -{\it b1}-{\it b2} \right ) {\omega }^{2}+ \left ( i{\it b1}\,a+i{\it b2}\,a \right ) \omega +{\it b2}\,{\it b1} \right ) {\it b2}\,a}}+{\frac {-{\it \_C2}\,a{\it b1}\,{\omega }^{4}+2\,i{\it \_C2}\,{a}^{2}{\it b1}\,{\omega }^{3}+ \left ( a{{\it b1}}^{2}+a{\it b1}\,{\it b2} \right ) {\it \_C2}\,{ \omega }^{2}+ \left ( -i{a}^{2}{{\it b1}}^{2}-i{a}^{2}{\it b1}\,{\it b2} \right ) {\it \_C2}\,\omega -{\it \_C2}\,a{{\it b1}}^{2}{\it b2}}{ \left ( {\omega }^{4}-2\,ia{\omega }^{3}+ \left ( -{\it b1}-{\it b2} \right ) {\omega }^{2}+ \left ( i{\it b1}\,a+i{\it b2}\,a \right ) \omega +{\it b2}\,{\it b1} \right ) {\it b2}\,a}} \right ) {{\rm e}^{{\it RootOf} \left ( {{\it \_Z}}^{4}+2\,a{{\it \_Z}}^{3}+ \left ( {\it b1}+{ \it b2} \right ) {{\it \_Z}}^{2}+ \left ( a{\it b1}+{\it b2}\,a \right ) {\it \_Z}+{\it b2}\,{\it b1},{\it index}=2 \right ) t}}+ \left ( {\frac { \left ( -{\it \_C1}\,{\omega }^{4}+2\,i{\it \_C1}\,a{\omega }^{3}+ \left ( {\it b1}+{\it b2} \right ) {\it \_C1}\,{\omega }^{2}+ \left ( -i{ \it b1}\,a-i{\it b2}\,a \right ) {\it \_C1}\,\omega -{\it \_C1}\,{\it b2 }\,{\it b1} \right ) \left ( {\it RootOf} \left ( {{\it \_Z}}^{4}+2\,a{{ \it \_Z}}^{3}+ \left ( {\it b1}+{\it b2} \right ) {{\it \_Z}}^{2}+ \left ( a{\it b1}+{\it b2}\,a \right ) {\it \_Z}+{\it b2}\,{\it b1},{ \it index}=1 \right ) \right ) ^{3}}{ \left ( {\omega }^{4}-2\,ia{\omega } ^{3}+ \left ( -{\it b1}-{\it b2} \right ) {\omega }^{2}+ \left ( i{\it b1} \,a+i{\it b2}\,a \right ) \omega +{\it b2}\,{\it b1} \right ) {\it b2}\,a }}+{\frac { \left ( -2\,{\it \_C1}\,a{\omega }^{4}+4\,i{\it \_C1}\,{a}^{ 2}{\omega }^{3}+ \left ( 2\,a{\it b1}+2\,{\it b2}\,a \right ) {\it \_C1} \,{\omega }^{2}+ \left ( -2\,i{a}^{2}{\it b1}-2\,i{a}^{2}{\it b2} \right ) {\it \_C1}\,\omega -2\,{\it \_C1}\,a{\it b1}\,{\it b2} \right ) \left ( {\it RootOf} \left ( {{\it \_Z}}^{4}+2\,a{{\it \_Z}}^{ 3}+ \left ( {\it b1}+{\it b2} \right ) {{\it \_Z}}^{2}+ \left ( a{\it b1} +{\it b2}\,a \right ) {\it \_Z}+{\it b2}\,{\it b1},{\it index}=1 \right ) \right ) ^{2}}{ \left ( {\omega }^{4}-2\,ia{\omega }^{3}+ \left ( -{\it b1}-{\it b2} \right ) {\omega }^{2}+ \left ( i{\it b1}\,a+i {\it b2}\,a \right ) \omega +{\it b2}\,{\it b1} \right ) {\it b2}\,a}}+{ \frac { \left ( -{\it \_C1}\,{\it b1}\,{\omega }^{4}+2\,i{\it \_C1}\,a{ \it b1}\,{\omega }^{3}+ \left ( {{\it b1}}^{2}+{\it b2}\,{\it b1} \right ) {\it \_C1}\,{\omega }^{2}+ \left ( -ia{{\it b1}}^{2}-ia{\it b1} \,{\it b2} \right ) {\it \_C1}\,\omega -{\it \_C1}\,{{\it b1}}^{2}{\it b2} \right ) {\it RootOf} \left ( {{\it \_Z}}^{4}+2\,a{{\it \_Z}}^{3}+ \left ( {\it b1}+{\it b2} \right ) {{\it \_Z}}^{2}+ \left ( a{\it b1}+{ \it b2}\,a \right ) {\it \_Z}+{\it b2}\,{\it b1},{\it index}=1 \right ) }{ \left ( {\omega }^{4}-2\,ia{\omega }^{3}+ \left ( -{\it b1}-{\it b2} \right ) {\omega }^{2}+ \left ( i{\it b1}\,a+i{\it b2}\,a \right ) \omega +{\it b2}\,{\it b1} \right ) {\it b2}\,a}}+{\frac {-{\it \_C1}\,a{\it b1}\,{\omega }^{4}+2\,i{\it \_C1}\,{a}^{2}{\it b1}\,{\omega }^{3}+ \left ( a{{\it b1}}^{2}+a{\it b1}\,{\it b2} \right ) {\it \_C1}\,{ \omega }^{2}+ \left ( -i{a}^{2}{{\it b1}}^{2}-i{a}^{2}{\it b1}\,{\it b2} \right ) {\it \_C1}\,\omega -{\it \_C1}\,a{{\it b1}}^{2}{\it b2}}{ \left ( {\omega }^{4}-2\,ia{\omega }^{3}+ \left ( -{\it b1}-{\it b2} \right ) {\omega }^{2}+ \left ( i{\it b1}\,a+i{\it b2}\,a \right ) \omega +{\it b2}\,{\it b1} \right ) {\it b2}\,a}} \right ) {{\rm e}^{{\it RootOf} \left ( {{\it \_Z}}^{4}+2\,a{{\it \_Z}}^{3}+ \left ( {\it b1}+{ \it b2} \right ) {{\it \_Z}}^{2}+ \left ( a{\it b1}+{\it b2}\,a \right ) {\it \_Z}+{\it b2}\,{\it b1},{\it index}=1 \right ) t}}+{\frac { \left ( -a{\it b2}\,{\it c2}\,{\omega }^{2}+ \left ( i{a}^{2}{\it b2}\,{ \it c1}+i{a}^{2}{\it b2}\,{\it c2} \right ) \omega +a{\it b1}\,{\it b2} \,{\it c2} \right ) {{\rm e}^{i\omega \,t}}}{ \left ( {\omega }^{4}-2\,ia{ \omega }^{3}+ \left ( -{\it b1}-{\it b2} \right ) {\omega }^{2}+ \left ( i{ \it b1}\,a+i{\it b2}\,a \right ) \omega +{\it b2}\,{\it b1} \right ) { \it b2}\,a}} \right \} \right \} \]