\[ \left \{\text {x1}'(t)=a \text {x2}(t)+b \text {x3}(t) \cos (c t)+b \text {x4}(t) \sin (c t),\text {x2}'(t)=-a \text {x1}(t)+b \text {x3}(t) \sin (c t)-b \text {x4}(t) \cos (c t),\text {x3}'(t)=a \text {x4}(t)-b \text {x1}(t) \cos (c t)-b \text {x2}(t) \sin (c t),\text {x4}'(t)=-a \text {x3}(t)-b \text {x1}(t) \sin (c t)+b \text {x2}(t) \cos (c t)\right \} \] ✗ Mathematica : cpu = 0.0880882 (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 = 2.739 (sec), leaf count = 11400
\[ \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}\,a-c{\it \_C3} \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}\,a+{\it \_C4}\,c \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}{8\,c \left ( {a}^{2}+ca+{b}^{2} \right ) } \left ( {\it \_C3}\, \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{{\frac {3}{2}}}{{\rm e}^{-{\frac {t}{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}}}}-{\it \_C4}\, \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{{\frac {3}{2}}}{{\rm e}^{{\frac {t}{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}}}}+4\,{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{\it \_C3}\,{a}^{2}+4\,{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{\it \_C3}\,ac+4\,{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{\it \_C3}\,{b}^{2}+4\,{c}^{2}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{\it \_C3}-4\,{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{\it \_C4}\,{a}^{2}-4\,{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{\it \_C4}\,ac-4\,{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{\it \_C4}\,{b}^{2}-4\,{c}^{2}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{\it \_C4}+{\it \_C1}\, \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{{\frac {3}{2}}}{{\rm e}^{-{\frac {t}{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}}}}-{\it \_C2}\, \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{{\frac {3}{2}}}{{\rm e}^{{\frac {t}{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}}}}+4\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,{a}^{2}+4\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,ac+4\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,{b}^{2}+4\,{c}^{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}-4\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,{a}^{2}-4\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,ac-4\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,{b}^{2}-4\,{c}^{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2} \right ) },{\it x3} \left ( t \right ) ={\frac {1}{8\,bc \left ( {a}^{2}+ca+{b}^{2} \right ) } \left ( \cos \left ( ct \right ) \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{{\frac {3}{2}}}{{\rm e}^{-{\frac {t}{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}}}}{\it \_C1}\,a+4\,{b}^{2}\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,\sin \left ( ct \right ) +4\,{b}^{2}\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,\sin \left ( ct \right ) +4\,{b}^{2}{c}^{2}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C4}\,\sin \left ( ct \right ) +4\,{b}^{2}{c}^{2}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,\sin \left ( ct \right ) +4\,{b}^{2}{c}^{2}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C3}\,\sin \left ( ct \right ) +4\,{b}^{2}{c}^{2}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,\sin \left ( ct \right ) -4\,{b}^{2}\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C4}\,\sin \left ( ct \right ) -4\,{b}^{2}\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C3}\,\sin \left ( ct \right ) +8\,\sin \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,{a}^{3}c+12\,{c}^{2}\sin \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,{a}^{2}+4\,{c}^{3}\sin \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,a+8\,\sin \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C3}\,{a}^{3}c+12\,{c}^{2}\sin \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C3}\,{a}^{2}+4\,{c}^{3}\sin \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C3}\,a+8\,\sin \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C4}\,{a}^{3}c+12\,{c}^{2}\sin \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C4}\,{a}^{2}+4\,{c}^{3}\sin \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C4}\,a+4\,\sin \left ( ct \right ) \sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,{a}^{2}-4\,\sin \left ( ct \right ) \sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C4}\,{a}^{2}-4\,\sin \left ( ct \right ) \sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C3}\,{a}^{2}-\cos \left ( ct \right ) {{\rm e}^{{\frac {t}{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}}}} \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{{\frac {3}{2}}}{\it \_C4}\,a-\cos \left ( ct \right ) \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{{\frac {3}{2}}}{{\rm e}^{{\frac {t}{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}}}}{\it \_C2}\,a+4\,\cos \left ( ct \right ) \sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,{a}^{3}-4\,\cos \left ( ct \right ) \sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,{a}^{3}-4\,\cos \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{\it \_C4}\,{a}^{3}+8\,\sin \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,{a}^{3}c+12\,{c}^{2}\sin \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,{a}^{2}+4\,{c}^{3}\sin \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,a+\cos \left ( ct \right ) {{\rm e}^{-{\frac {t}{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}}}} \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{{\frac {3}{2}}}{\it \_C3}\,a+4\,\sin \left ( ct \right ) \sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,{a}^{2}+4\,\cos \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{\it \_C3}\,{a}^{3}+4\,{b}^{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,a\cos \left ( ct \right ) +4\,{b}^{2}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{\it \_C3}\,a\cos \left ( ct \right ) +4\,{\it \_C4}\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{b}^{2}c\cos \left ( ct \right ) -4\,{\it \_C3}\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{b}^{2}c\cos \left ( ct \right ) +4\,{\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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{b}^{2}c\cos \left ( ct \right ) -4\,{\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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{b}^{2}c\cos \left ( ct \right ) -4\,{b}^{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,a\cos \left ( ct \right ) -4\,{b}^{2}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{\it \_C4}\,a\cos \left ( ct \right ) +8\,{b}^{2}\sin \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,ac+8\,{b}^{2}\sin \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,ac+8\,{b}^{2}\sin \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C3}\,ac+8\,{b}^{2}\sin \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C4}\,ac+4\,\sin \left ( ct \right ) \sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,ac+4\,\sin \left ( ct \right ) \sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,ac-4\,\sin \left ( ct \right ) \sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C3}\,ac-4\,\sin \left ( ct \right ) \sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C4}\,ac \right ) },{\it x4} \left ( t \right ) =-{\frac {1}{8\,bc \left ( {a}^{2}+ca+{b}^{2} \right ) } \left ( 4\,\cos \left ( ct \right ) \sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,{a}^{2}+4\,\sin \left ( ct \right ) \sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,{a}^{3}+8\,\cos \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,{a}^{3}c+12\,{c}^{2}\cos \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,{a}^{2}+4\,{c}^{3}\cos \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,a+8\,\cos \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,{a}^{3}c+4\,\cos \left ( ct \right ) \sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,ac+4\,\cos \left ( ct \right ) \sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,ac-4\,\cos \left ( ct \right ) \sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C3}\,ac-4\,\cos \left ( ct \right ) \sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C4}\,ac-4\,\cos \left ( ct \right ) \sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C4}\,{a}^{2}+\sin \left ( ct \right ) {{\rm e}^{{\frac {t}{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}}}} \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{{\frac {3}{2}}}{\it \_C4}\,a+4\,\sin \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{\it \_C4}\,{a}^{3}-\sin \left ( ct \right ) \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{{\frac {3}{2}}}{{\rm e}^{-{\frac {t}{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}}}}{\it \_C1}\,a+4\,{b}^{2}{c}^{2}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C4}\,\cos \left ( ct \right ) -4\,{b}^{2}\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C3}\,\cos \left ( ct \right ) -4\,{b}^{2}\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C4}\,\cos \left ( ct \right ) +4\,{b}^{2}{c}^{2}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,\cos \left ( ct \right ) +4\,{b}^{2}{c}^{2}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,\cos \left ( ct \right ) +4\,{b}^{2}{c}^{2}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C3}\,\cos \left ( ct \right ) +4\,{b}^{2}\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,\cos \left ( ct \right ) +4\,{b}^{2}\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,\cos \left ( ct \right ) +12\,{c}^{2}\cos \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,{a}^{2}+4\,{c}^{3}\cos \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,a+8\,\cos \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C3}\,{a}^{3}c+12\,{c}^{2}\cos \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C3}\,{a}^{2}+4\,{c}^{3}\cos \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C3}\,a+8\,\cos \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C4}\,{a}^{3}c+12\,{c}^{2}\cos \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C4}\,{a}^{2}+4\,{c}^{3}\cos \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C4}\,a+8\,{b}^{2}\cos \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,ac+8\,{b}^{2}\cos \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,ac+8\,{b}^{2}\cos \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C3}\,ac+8\,{b}^{2}\cos \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C4}\,ac+4\,\cos \left ( ct \right ) \sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,{a}^{2}-\sin \left ( ct \right ) {{\rm e}^{-{\frac {t}{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}}}} \left ( -4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{{\frac {3}{2}}}{\it \_C3}\,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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}} \right ) ^{{\frac {3}{2}}}{{\rm e}^{{\frac {t}{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}}}}{\it \_C2}\,a-4\,\sin \left ( ct \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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{\it \_C3}\,{a}^{3}-4\,\sin \left ( ct \right ) \sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,{a}^{3}-4\,\cos \left ( ct \right ) \sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C3}\,{a}^{2}+4\,{\it \_C3}\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{b}^{2}c\sin \left ( ct \right ) -4\,{\it \_C4}\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{b}^{2}c\sin \left ( ct \right ) -4\,{b}^{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C1}\,a\sin \left ( ct \right ) +4\,{b}^{2}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{\it \_C4}\,a\sin \left ( ct \right ) +4\,{b}^{2}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{{\rm e}^{1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}{\it \_C2}\,a\sin \left ( ct \right ) -4\,{b}^{2}{{\rm e}^{-1/2\,\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}+2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{\it \_C3}\,a\sin \left ( ct \right ) -4\,{\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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{b}^{2}c\sin \left ( ct \right ) +4\,{\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\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}t}}\sqrt {-4\,{a}^{2}-4\,ca-4\,{b}^{2}-2\,{c}^{2}-2\,\sqrt {4\,{a}^{2}{c}^{2}+4\,{c}^{3}a+4\,{b}^{2}{c}^{2}+{c}^{4}}}{b}^{2}c\sin \left ( ct \right ) \right ) } \right \} \right \} \]