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人気のある微分積分 >

(dP)/(dt)=0.2P(1-P/(1000))-32

解

\frac{d P}{d t} = 0.2 P (1 - \frac{P}{1000}) - 32

解

P = - \frac{800 e ^{\frac{3 t - 3 c _{1}}{25}}}{1 - e ^{\frac{3 ( - c _{1} + t )}{25}}} + \frac{200}{1 - e ^{\frac{3 ( - c _{1} + t )}{25}}}

解答ステップ

\frac{d P}{d t} = 0.2 P (1 - \frac{P}{1000}) - 32

代用

\frac{d P}{d t}

と

P^{'} (t)

P^{^{'}} (t) = 0.2 P (1 - \frac{P}{1000}) - 32

t

を解く

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

(\frac{1}{t ^{^{^{'}}}}) = 0.2 P (1 - \frac{P}{1000}) - 32

分離する

t^{^{'}} : t^{^{'}} = \frac{5000}{1000 P - P ^{2} - 160000}

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

であれば

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

t = \int \frac{5000}{1000 P - P ^{2} - 160000} d P

\int \frac{5000}{1000 P - P ^{2} - 160000} d P = \frac{25}{3} (ln (\frac{P - 500}{300} + 1) - ln (\frac{P - 500}{300} - 1)) + c_{1}

t = \frac{25}{3} (ln (\frac{P - 500}{300} + 1) - ln (\frac{P - 500}{300} - 1)) + c_{1}

簡素化

t = \frac{25}{3} ln (\frac{P - 200}{P - 800}) + c_{1}

分離する

P : P = - \frac{800 e ^{\frac{3 t - 3 c _{1}}{25}}}{1 - e ^{\frac{3 ( - c _{1} + t )}{25}}} + \frac{200}{1 - e ^{\frac{3 ( - c _{1} + t )}{25}}}

P = - \frac{800 e ^{\frac{3 t - 3 c _{1}}{25}}}{1 - e ^{\frac{3 ( - c _{1} + t )}{25}}} + \frac{200}{1 - e ^{\frac{3 ( - c _{1} + t )}{25}}}

グラフ

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