inverselaplace (5s-2)/(7s^2-s-11)

解

逆ラプラス

\frac{5 s - 2}{7 s ^{2} - s - 11}

\frac{5}{7} e^{\frac{t}{14}} cosh (\frac{309 t}{14}) - \frac{23 e ^{\frac{t}{14}} sinh ( \frac{309 t}{14} )}{7 309}

解答ステップ

L^{- 1} {\frac{5 s - 2}{7 s ^{2} - s - 11}}

拡張

\frac{5 s - 2}{7 s ^{2} - s - 11} : \frac{5}{7} \cdot \frac{s - \frac{1}{14}}{( s - \frac{1}{14} ) ^{2} - \frac{309}{196}} - \frac{23}{98} \cdot \frac{1}{( s - \frac{1}{14} ) ^{2} - \frac{309}{196}}

= L^{- 1} {\frac{5}{7} \cdot \frac{s - \frac{1}{14}}{( s - \frac{1}{14} ) ^{2} - \frac{309}{196}} - \frac{23}{98} \cdot \frac{1}{( s - \frac{1}{14} ) ^{2} - \frac{309}{196}}}

逆ラプラス変換の線形性を使用する:

関数

f (s), g (s)

と定数

a, b : L^{- 1} {a \cdot f (s) + b \cdot g (s)} = a \cdot L^{- 1} {f (s)} + b \cdot L^{- 1} {g (s)}

= \frac{5}{7} L^{- 1} {\frac{s - \frac{1}{14}}{( s - \frac{1}{14} ) ^{2} - \frac{309}{196}}} - \frac{23}{98} L^{- 1} {\frac{1}{( s - \frac{1}{14} ) ^{2} - \frac{309}{196}}}

L^{- 1} {\frac{s - \frac{1}{14}}{( s - \frac{1}{14} ) ^{2} - \frac{309}{196}}} : e^{\frac{t}{14}} cosh (\frac{309 t}{14})

L^{- 1} {\frac{1}{( s - \frac{1}{14} ) ^{2} - \frac{309}{196}}} : e^{\frac{t}{14}} \frac{14}{309} sinh (\frac{309 t}{14})

= \frac{5}{7} e^{\frac{t}{14}} cosh (\frac{309 t}{14}) - \frac{23}{98} e^{\frac{t}{14}} \frac{14}{309} sinh (\frac{309 t}{14})

改良

\frac{5}{7} e^{\frac{t}{14}} cosh (\frac{309 t}{14}) - \frac{23}{98} e^{\frac{t}{14}} \frac{14}{309} sinh (\frac{309 t}{14}) : \frac{5}{7} e^{\frac{t}{14}} cosh (\frac{309 t}{14}) - \frac{23 e ^{\frac{t}{14}} sinh ( \frac{309 t}{14} )}{7 309}

= \frac{5}{7} e^{\frac{t}{14}} cosh (\frac{309 t}{14}) - \frac{23 e ^{\frac{t}{14}} sinh ( \frac{309 t}{14} )}{7 309}