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受欢迎的微积分 >

(dy)/(y(A-y))= k/A dx

解答

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

解答

y = - \frac{A e ^{c_{1} k}}{e ^{k x} - e ^{c_{1} k}}

求解步骤

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

令

x

为因变量。除以

d y

:

\frac{1}{y ( A - y )} = \frac{k}{A} \cdot \frac{d x}{d y}

用

x^{^{'}}

替代

\frac{d x}{d y}

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

整理

x^{^{'}} : x^{^{'}} = \frac{A}{k y ( A - y )}

若

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

则

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

x = \int \frac{A}{k y ( A - y )} d y

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

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

化简

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

整理

y : y = - \frac{A e ^{c_{1} k}}{e ^{k x} - e ^{c_{1} k}}

y = - \frac{A e ^{c_{1} k}}{e ^{k x} - e ^{c_{1} k}}

作图

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