y^{'''}-y^{''}+y^'-y=sec(t)

解

y^{^{'''}} - y^{^{''}} + y^{^{'}} - y = sec (t)

y = c_{1} e^{t} + c_{2} cos (t) + c_{3} sin (t) + e^{t} \cdot \int \frac{sec ( t )}{2 e ^{t}} d t + \frac{1}{2} cos (t) (- t - ln (cos (t))) - \frac{1}{2} sin (t) (- ln (cos (t)) + t)

解答ステップ

y^{'''} (t) - y^{''} (t) + y^{'} (t) - y = sec (t)

線形 ODE を解く:

y = c_{1} e^{t} + c_{2} cos (t) + c_{3} sin (t) + e^{t} \cdot \int \frac{sec ( t )}{2 e ^{t}} d t + \frac{1}{2} cos (t) (- t - ln (cos (t))) - \frac{1}{2} sin (t) (- ln (cos (t)) + t)

y = c_{1} e^{t} + c_{2} cos (t) + c_{3} sin (t) + e^{t} \cdot \int \frac{sec ( t )}{2 e ^{t}} d t + \frac{1}{2} cos (t) (- t - ln (cos (t))) - \frac{1}{2} sin (t) (- ln (cos (t)) + t)