2.1905   ODE No. 1905

1. Problem in Latex
2. Mathematica input
3. Maple input

\[ \left \{x'(t)=h(t) y(t)-g(t) z(t),y'(t)=f(t) z(t)-h(t) x(t),z'(t)=g(t) x(t)-f(t) y(t)\right \} \] ✗ Mathematica : cpu = 0.00693789 (sec), leaf count = 0 , could not solve

DSolve[{Derivative[1][x][t] == h[t]*y[t] - g[t]*z[t], Derivative[1][y][t] == -(h[t]*x[t]) + f[t]*z[t], Derivative[1][z][t] == g[t]*x[t] - f[t]*y[t]}, {x[t], y[t], z[t]}, t]

✗ Maple : cpu = 0. (sec), leaf count = 0 , result contains DESol

\[ \left \{ \left \{ x \left ( t \right ) ={\it DESol} \left ( \left \{ {\frac {{\rm d}^{3}}{{\rm d}{t}^{3}}}{\it \_Y} \left ( t \right ) + \left ( -2\,{\frac {h \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) f \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}-2\,{\frac {g \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) \right ) f \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+{\frac {h \left ( t \right ) {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}g \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}-{\frac {g \left ( t \right ) {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}h \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}-{\frac { \left ( h \left ( t \right ) \right ) ^{2}{\frac {\rm d}{{\rm d}t}}f \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}-{\frac { \left ( g \left ( t \right ) \right ) ^{2}{\frac {\rm d}{{\rm d}t}}f \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }} \right ) {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}{\it \_Y} \left ( t \right ) + \left ( -3\,{\frac {h \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) \right ) \left ( f \left ( t \right ) \right ) ^{2}}{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}-{\frac {h \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) \right ) \left ( g \left ( t \right ) \right ) ^{2}}{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+{\frac {h \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}-{\frac {g \left ( t \right ) \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}g \left ( t \right ) \right ) f \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}-{\frac {h \left ( t \right ) \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}h \left ( t \right ) \right ) f \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+{\frac { \left ( {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) \right ) g \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+2\,{\frac { \left ( h \left ( t \right ) \right ) ^{2} \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+{\frac { \left ( h \left ( t \right ) \right ) ^{2} \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+3\,{\frac { \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) \left ( f \left ( t \right ) \right ) ^{2}}{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+{\frac { \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}h \left ( t \right ) \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+2\,{\frac { \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) ^{2}f \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+2\,{\frac { \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) \right ) ^{2}f \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}-{\frac { \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}g \left ( t \right ) \right ) {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+{\frac { \left ( h \left ( t \right ) \right ) ^{2} \left ( f \left ( t \right ) \right ) ^{3}}{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+{\frac { \left ( g \left ( t \right ) \right ) ^{2} \left ( f \left ( t \right ) \right ) ^{3}}{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+{\frac { \left ( g \left ( t \right ) \right ) ^{4}f \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+{\frac { \left ( g \left ( t \right ) \right ) ^{3}{\frac {\rm d}{{\rm d}t}}h \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}-{\frac { \left ( h \left ( t \right ) \right ) ^{3}{\frac {\rm d}{{\rm d}t}}g \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+{\frac { \left ( h \left ( t \right ) \right ) ^{4}f \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }} \right ) {\frac {\rm d}{{\rm d}t}}{\it \_Y} \left ( t \right ) + \left ( {\frac { \left ( g \left ( t \right ) \right ) ^{2} \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) h \left ( t \right ) f \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+{\frac {g \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) \right ) \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+{\frac { \left ( g \left ( t \right ) \right ) ^{2} \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}g \left ( t \right ) \right ) h \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}-{\frac {g \left ( t \right ) \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}h \left ( t \right ) \right ) \left ( h \left ( t \right ) \right ) ^{2}}{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}-3\,{\frac {g \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) \right ) ^{2}h \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}-2\,{\frac { \left ( g \left ( t \right ) \right ) ^{2} \left ( {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) \right ) \left ( h \left ( t \right ) \right ) ^{2}}{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+{\frac { \left ( g \left ( t \right ) \right ) ^{3} \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) \right ) f \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+3\,{\frac {g \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) ^{2}h \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}-3\,{\frac { \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) \right ) \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) \left ( h \left ( t \right ) \right ) ^{2}}{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+{\frac { \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) \left ( h \left ( t \right ) \right ) ^{3}f \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+3\,{\frac { \left ( g \left ( t \right ) \right ) ^{2} \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) \right ) {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}-{\frac { \left ( {\frac {\rm d}{{\rm d}t}}f \left ( t \right ) \right ) \left ( h \left ( t \right ) \right ) ^{4}}{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}-{\frac { \left ( g \left ( t \right ) \right ) ^{4}{\frac {\rm d}{{\rm d}t}}f \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}+{\frac { \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}g \left ( t \right ) \right ) \left ( h \left ( t \right ) \right ) ^{3}}{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }}-{\frac { \left ( g \left ( t \right ) \right ) ^{3}{\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}h \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }} \right ) {\it \_Y} \left ( t \right ) \right \} , \left \{ {\it \_Y} \left ( t \right ) \right \} \right ) ,y \left ( t \right ) ={\frac { \left ( h \left ( t \right ) \right ) ^{2}x \left ( t \right ) g \left ( t \right ) +x \left ( t \right ) \left ( g \left ( t \right ) \right ) ^{3}+h \left ( t \right ) \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) f \left ( t \right ) + \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) g \left ( t \right ) - \left ( {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) \right ) {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }},z \left ( t \right ) ={\frac { \left ( h \left ( t \right ) \right ) ^{3}x \left ( t \right ) +h \left ( t \right ) x \left ( t \right ) \left ( g \left ( t \right ) \right ) ^{2}- \left ( {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) \right ) g \left ( t \right ) f \left ( t \right ) + \left ( {\frac {{\rm d}^{2}}{{\rm d}{t}^{2}}}x \left ( t \right ) \right ) h \left ( t \right ) - \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) {\frac {\rm d}{{\rm d}t}}x \left ( t \right ) }{ \left ( h \left ( t \right ) \right ) ^{2}f \left ( t \right ) + \left ( g \left ( t \right ) \right ) ^{2}f \left ( t \right ) -h \left ( t \right ) {\frac {\rm d}{{\rm d}t}}g \left ( t \right ) + \left ( {\frac {\rm d}{{\rm d}t}}h \left ( t \right ) \right ) g \left ( t \right ) }} \right \} \right \} \]