\[ \left \{\text {x1}'(t)=a \text {x2}(t)+b \text {x3}(t) \cos (c t)+b \text {x4}(t) \sin (c t),\text {x2}'(t)=-a \text {x1}(t)+b \text {x3}(t) \sin (c t)-b \text {x4}(t) \cos (c t),\text {x3}'(t)=a \text {x4}(t)-b \text {x1}(t) \cos (c t)-b \text {x2}(t) \sin (c t),\text {x4}'(t)=-a \text {x3}(t)-b \text {x1}(t) \sin (c t)+b \text {x2}(t) \cos (c t)\right \} \] ✗ Mathematica : cpu = 0.0112033 (sec), leaf count = 0 , could not solve
DSolve[{Derivative[1][x1][t] == a*x2[t] + b*Cos[c*t]*x3[t] + b*Sin[c*t]*x4[t], Derivative[1][x2][t] == -(a*x1[t]) + b*Sin[c*t]*x3[t] - b*Cos[c*t]*x4[t], Derivative[1][x3][t] == -(b*Cos[c*t]*x1[t]) - b*Sin[c*t]*x2[t] + a*x4[t], Derivative[1][x4][t] == -(b*Sin[c*t]*x1[t]) + b*Cos[c*t]*x2[t] - a*x3[t]}, {x[t], y[t], z[t]}, t]
✓ Maple : cpu = 2.47 (sec), leaf count = 2956
\[ \left \{ \left \{ {\it x1} \left ( t \right ) ={\it \_C2}+{\it \_C3}\,\sin \left ( ct \right ) +{\it \_C4}\,\cos \left ( ct \right ) ,{\it x2} \left ( t \right ) =-\cos \left ( ct \right ) {\it \_C3}+\sin \left ( ct \right ) {\it \_C4}+{\it \_C1},{\it x3} \left ( t \right ) ={\frac {b \left ( \cos \left ( ct \right ) {\it \_C1}\,a-\sin \left ( ct \right ) {\it \_C2}\,a-{\it \_C3}\, \left ( a+c \right ) \right ) }{a \left ( a+c \right ) }},{\it x4} \left ( t \right ) ={\frac {b \left ( \cos \left ( ct \right ) {\it \_C2}\,a+\sin \left ( ct \right ) {\it \_C1}\,a+{\it \_C4}\, \left ( a+c \right ) \right ) }{a \left ( a+c \right ) }} \right \} , \left \{ {\it x1} \left ( t \right ) ={\it \_C1}\,{{\rm e}^{-{\frac {t}{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {{c}^{2} \left ( 4\,{a}^{2}+4\,ca+4\,{b}^{2}+{c}^{2} \right ) }}}}}+{\it \_C2}\,{{\rm e}^{{\frac {t}{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {{c}^{2} \left ( 4\,{a}^{2}+4\,ca+4\,{b}^{2}+{c}^{2} \right ) }}}}}+{\it \_C3}\,{{\rm e}^{-{\frac {t}{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {{c}^{2} \left ( 4\,{a}^{2}+4\,ca+4\,{b}^{2}+{c}^{2} \right ) }}}}}+{\it \_C4}\,{{\rm e}^{{\frac {t}{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {{c}^{2} \left ( 4\,{a}^{2}+4\,ca+4\,{b}^{2}+{c}^{2} \right ) }}}}},{\it x2} \left ( t \right ) ={\frac {1}{ \left ( 8\,{a}^{2}+8\,ca+8\,{b}^{2} \right ) c} \left ( -4\,{\it \_C1}\, \left ( 1/4\, \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{3/2}+\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}} \left ( {a}^{2}+ca+{b}^{2}+{c}^{2} \right ) \right ) {{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}+4\, \left ( 1/4\, \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{3/2}+\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}} \left ( {a}^{2}+ca+{b}^{2}+{c}^{2} \right ) \right ) {\it \_C2}\,{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}-4\, \left ( {\it \_C3}\,{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}-{\it \_C4}\,{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}} \right ) \left ( 1/4\, \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{3/2}+ \left ( {a}^{2}+ca+{b}^{2}+{c}^{2} \right ) \sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}} \right ) \right ) },{\it x3} \left ( t \right ) ={\frac {1}{8\,bc \left ( {a}^{2}+ca+{b}^{2} \right ) } \left ( 8\, \left ( 1/8\,\cos \left ( ct \right ) \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{3/2}a+1/2\,\cos \left ( ct \right ) \left ( {a}^{3}+a{b}^{2}-{b}^{2}c \right ) \sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}}+ \left ( {a}^{2}+ca+{b}^{2} \right ) \left ( 1/2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}+c \left ( a+c/2 \right ) \right ) \sin \left ( ct \right ) \right ) {\it \_C1}\,{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}+8\,{\it \_C2}\, \left ( -1/8\,\cos \left ( ct \right ) \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{3/2}a-1/2\,\cos \left ( ct \right ) \left ( {a}^{3}+a{b}^{2}-{b}^{2}c \right ) \sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}}+ \left ( {a}^{2}+ca+{b}^{2} \right ) \left ( 1/2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}+c \left ( a+c/2 \right ) \right ) \sin \left ( ct \right ) \right ) {{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}+8\,{\it \_C3}\, \left ( 1/8\,a \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{3/2}\cos \left ( ct \right ) +1/2\,\cos \left ( ct \right ) \left ( {a}^{3}+a{b}^{2}-{b}^{2}c \right ) \sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}}+ \left ( {a}^{2}+ca+{b}^{2} \right ) \left ( -1/2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}+c \left ( a+c/2 \right ) \right ) \sin \left ( ct \right ) \right ) {{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}+8\, \left ( -1/8\,a \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{3/2}\cos \left ( ct \right ) -1/2\,\cos \left ( ct \right ) \left ( {a}^{3}+a{b}^{2}-{b}^{2}c \right ) \sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}}+ \left ( {a}^{2}+ca+{b}^{2} \right ) \left ( -1/2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}+c \left ( a+c/2 \right ) \right ) \sin \left ( ct \right ) \right ) {\it \_C4}\,{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}} \right ) },{\it x4} \left ( t \right ) ={\frac {1}{8\,bc \left ( {a}^{2}+ca+{b}^{2} \right ) } \left ( 4\,{\it \_C1}\, \left ( 1/4\,a\sin \left ( ct \right ) \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{3/2}+\sin \left ( ct \right ) \left ( {a}^{3}+a{b}^{2}-{b}^{2}c \right ) \sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}}-2\, \left ( {a}^{2}+ca+{b}^{2} \right ) \cos \left ( ct \right ) \left ( 1/2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}+c \left ( a+c/2 \right ) \right ) \right ) {{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}-4\, \left ( 1/4\,a\sin \left ( ct \right ) \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{3/2}+\sin \left ( ct \right ) \left ( {a}^{3}+a{b}^{2}-{b}^{2}c \right ) \sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}}+2\, \left ( {a}^{2}+ca+{b}^{2} \right ) \cos \left ( ct \right ) \left ( 1/2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}+c \left ( a+c/2 \right ) \right ) \right ) {\it \_C2}\,{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}+4\,{\it \_C3}\, \left ( 1/4\,a\sin \left ( ct \right ) \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{3/2}+\sin \left ( ct \right ) \left ( {a}^{3}+a{b}^{2}-{b}^{2}c \right ) \sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}}-2\, \left ( {a}^{2}+ca+{b}^{2} \right ) \cos \left ( ct \right ) \left ( -1/2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}+c \left ( a+c/2 \right ) \right ) \right ) {{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}-4\, \left ( 1/4\,a\sin \left ( ct \right ) \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{3/2}+\sin \left ( ct \right ) \left ( {a}^{3}+a{b}^{2}-{b}^{2}c \right ) \sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}}+2\, \left ( {a}^{2}+ca+{b}^{2} \right ) \cos \left ( ct \right ) \left ( -1/2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}+c \left ( a+c/2 \right ) \right ) \right ) {\it \_C4}\,{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,a{c}^{3}+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}} \right ) } \right \} \right \} \]