y^'=sqrt(y)(1-sqrt(y))

Lösung

y^{^{'}} = y (1 - y)

y = e^{- t + c_{1}} + 2 e^{\frac{- t + c _{1}}{2}} + 1

Schritte zur Lösung

y^{'} (t) = y (1 - y)

Löse nach

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

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

Isoliere

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

Wenn

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

dann

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

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

\int - \frac{1}{y ( y - 1 )} d y = - 2 ln (y - 1) + c_{1}

t = - 2 ln (y - 1) + c_{1}

Isoliere

y : y = e^{- t + c_{1}} + 2 e^{\frac{- t + c _{1}}{2}} + 1

y = e^{- t + c_{1}} + 2 e^{\frac{- t + c _{1}}{2}} + 1

x \to - 4 lim (8 x^{2} + 1)

t a y l or - 2 sin (4 x)

d er i v a t i v e y = ln (x^{8})

\frac{d y}{d x} = x^{5} (y - 3), y (0) = 5

\int (x + 1) (x + 2)^{6} d x