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人気のある微分積分 >

(dv)/(dt)=v^2+5v+1

解

\frac{d v}{d t} = v^{2} + 5 v + 1

解

v = \frac{5 e ^{21 c_{1}}}{2 ( e ^{21 t} - e ^{21 c_{1}} )} + \frac{21 e ^{21 c_{1}}}{2 ( e ^{21 t} - e ^{21 c_{1}} )} - \frac{5 e ^{21 t}}{2 ( e ^{21 t} - e ^{21 c_{1}} )} + \frac{21 e ^{21 t}}{2 ( e ^{21 t} - e ^{21 c_{1}} )}

解答ステップ

\frac{d v}{d t} = v^{2} + 5 v + 1

代用

\frac{d v}{d t}

と

v^{'} (t)

v^{^{'}} (t) = v^{2} + 5 v + 1

t

を解く

v^{'} (t) = \frac{d v}{d t} = \frac{1}{\frac{d t}{d v}} = \frac{1}{t ^{^{'}}}

(\frac{1}{t ^{^{^{'}}}}) = v^{2} + 5 v + 1

分離する

t^{^{'}} : t^{^{'}} = \frac{1}{v ^{2} + 5 v + 1}

f^{^{'}} (x) = g (x)

であれば

f (x) = \int g (x) d x

t = \int \frac{1}{v ^{2} + 5 v + 1} d v

\int \frac{1}{v ^{2} + 5 v + 1} d v = - \frac{1}{21} (ln (\frac{2 v + 5}{21} + 1) - ln (\frac{2 v + 5}{21} - 1)) + c_{1}

t = - \frac{1}{21} (ln (\frac{2 v + 5}{21} + 1) - ln (\frac{2 v + 5}{21} - 1)) + c_{1}

分離する

v : v = \frac{5 e ^{21 c_{1}}}{2 ( e ^{21 t} - e ^{21 c_{1}} )} + \frac{21 e ^{21 c_{1}}}{2 ( e ^{21 t} - e ^{21 c_{1}} )} - \frac{5 e ^{21 t}}{2 ( e ^{21 t} - e ^{21 c_{1}} )} + \frac{21 e ^{21 t}}{2 ( e ^{21 t} - e ^{21 c_{1}} )}

v = \frac{5 e ^{21 c_{1}}}{2 ( e ^{21 t} - e ^{21 c_{1}} )} + \frac{21 e ^{21 c_{1}}}{2 ( e ^{21 t} - e ^{21 c_{1}} )} - \frac{5 e ^{21 t}}{2 ( e ^{21 t} - e ^{21 c_{1}} )} + \frac{21 e ^{21 t}}{2 ( e ^{21 t} - e ^{21 c_{1}} )}

グラフ

人気の例

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\int tan (x) sec^{2} (x) d x

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