(dy)/(dx)=k(10-y)(5-y)

Lösung

\frac{d y}{d x} = k (10 - y) (5 - y)

y = \frac{5 e ^{5 c_{1} k}}{- e ^{5 k x} + e ^{5 c_{1} k}} - \frac{10 e ^{5 k x}}{- e ^{5 k x} + e ^{5 c_{1} k}}

Schritte zur Lösung

\frac{d y}{d x} = k (10 - y) (5 - y)

Ersetze

\frac{d y}{d x}

mit

y^{'} (x)

y^{^{'}} (x) = k (10 - y) (5 - y)

Löse nach

x, y^{'} (x) = \frac{d y}{d x} = \frac{1}{\frac{d x}{d y}} = \frac{1}{x ^{^{'}}}

(\frac{1}{x ^{^{^{'}}}}) = k (10 - y) (5 - y)

Isoliere

x^{^{'}} : x^{^{'}} = \frac{1}{k ( 10 - y ) ( 5 - y )}

Wenn

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

dann

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

x = \int \frac{1}{k ( 10 - y ) ( 5 - y )} d y

\int \frac{1}{k ( 10 - y ) ( 5 - y )} d y = - \frac{1}{5 k} (- ln (10 - y) + ln (- y + 5)) + c_{1}

x = - \frac{1}{5 k} (- ln (10 - y) + ln (- y + 5)) + c_{1}

Vereinfache

x = - \frac{- ln ( 10 - y ) + ln ( - y + 5 )}{5 k} + c_{1}

Isoliere

y : y = \frac{5 e ^{5 c_{1} k}}{- e ^{5 k x} + e ^{5 c_{1} k}} - \frac{10 e ^{5 k x}}{- e ^{5 k x} + e ^{5 c_{1} k}}

y = \frac{5 e ^{5 c_{1} k}}{- e ^{5 k x} + e ^{5 c_{1} k}} - \frac{10 e ^{5 k x}}{- e ^{5 k x} + e ^{5 c_{1} k}}

\int x^{- 9} d x

\frac{d}{d x} (\frac{( x ^{2} + 2 x - 3 )}{( x + 1 ) ^{2}})

t an g e n t f (x) = x^{2} + 4 x - 1

\int 3 x^{\frac{1}{3}} d x

p a r i t y x - tan (x)