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

Lösung

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

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}}}

Schritte zur Lösung

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

Ersetze

\frac{d P}{d t}

mit

P^{'} (t)

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

Löse nach

t, 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

Isoliere

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

Wenn

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

dann

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}

Vereinfache

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

Isoliere

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}}}

\frac{d}{d x} (lo g_{7} (ln (x)))

\int e^{a r c c o s (x)} d x

\int 2 x - \frac{1}{x ^{2}} d x

\int 2^{x} + x^{2} d x

\frac{\partial}{\partial z} (yz - ln (x + z))