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人気のある微分積分 >

(dy)/(dt)=k(80-y)(50-y)

解

\frac{d y}{d t} = k (80 - y) (50 - y)

解

y = \frac{50 e ^{30 c_{1} k}}{- e ^{30 k t} + e ^{30 c_{1} k}} - \frac{80 e ^{30 k t}}{- e ^{30 k t} + e ^{30 c_{1} k}}

解答ステップ

\frac{d y}{d t} = k (80 - y) (50 - y)

代用

\frac{d y}{d t}

と

y^{'} (t)

y^{^{'}} (t) = k (80 - y) (50 - y)

t

を解く

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

(\frac{1}{t ^{^{^{'}}}}) = k (80 - y) (50 - y)

分離する

t^{^{'}} : t^{^{'}} = \frac{1}{k ( 80 - y ) ( 50 - y )}

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

であれば

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

t = \int \frac{1}{k ( 80 - y ) ( 50 - y )} d y

\int \frac{1}{k ( 80 - y ) ( 50 - y )} d y = - \frac{1}{30 k} (- ln (80 - y) + ln (- y + 50)) + c_{1}

t = - \frac{1}{30 k} (- ln (80 - y) + ln (- y + 50)) + c_{1}

簡素化

t = - \frac{- ln ( 80 - y ) + ln ( - y + 50 )}{30 k} + c_{1}

分離する

y : y = \frac{50 e ^{30 c_{1} k}}{- e ^{30 k t} + e ^{30 c_{1} k}} - \frac{80 e ^{30 k t}}{- e ^{30 k t} + e ^{30 c_{1} k}}

y = \frac{50 e ^{30 c_{1} k}}{- e ^{30 k t} + e ^{30 c_{1} k}} - \frac{80 e ^{30 k t}}{- e ^{30 k t} + e ^{30 c_{1} k}}

グラフ

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