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Beliebt Rechnen >

(dy)/(dt)=-2k(y^2-2)

Lösung

\frac{d y}{d t} = - 2 k (y^{2} - 2)

Lösung

y = \frac{2 e ^{42 c_{1} k}}{e ^{42 k t} - e ^{42 c_{1} k}} + \frac{2 e ^{42 k t}}{e ^{42 k t} - e ^{42 c_{1} k}}

Schritte zur Lösung

\frac{d y}{d t} = - 2 k (y^{2} - 2)

Ersetze

\frac{d y}{d t}

mit

y^{'} (t)

y^{^{'}} (t) = - 2 k (y^{2} - 2)

Löse nach

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

(\frac{1}{t ^{^{^{'}}}}) = - 2 k (y^{2} - 2)

Isoliere

t^{^{'}} : t^{^{'}} = - \frac{1}{2 k ( y ^{2} - 2 )}

Wenn

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

dann

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

t = \int - \frac{1}{2 k ( y ^{2} - 2 )} d y

\int - \frac{1}{2 k ( y ^{2} - 2 )} d y = \frac{1}{4 2 k} (ln (\frac{y}{2} + 1) - ln (\frac{y}{2} - 1)) + c_{1}

t = \frac{1}{4 2 k} (ln (\frac{y}{2} + 1) - ln (\frac{y}{2} - 1)) + c_{1}

Vereinfache

t = \frac{ln ( \frac{y}{2} + 1 ) - ln ( \frac{y}{2} - 1 )}{4 2 k} + c_{1}

Isoliere

y : y = \frac{2 e ^{42 c_{1} k}}{e ^{42 k t} - e ^{42 c_{1} k}} + \frac{2 e ^{42 k t}}{e ^{42 k t} - e ^{42 c_{1} k}}

y = \frac{2 e ^{42 c_{1} k}}{e ^{42 k t} - e ^{42 c_{1} k}} + \frac{2 e ^{42 k t}}{e ^{42 k t} - e ^{42 c_{1} k}}

Graph

Beliebte Beispiele

y^{^{'}} = 3 y e^{t}

\frac{y ^{^{'}}}{x} = 1 + y^{2}

(dy)/(dx)+ye^x=0

\frac{d y}{d x} + y e^{x} = 0

y^{^{'}} - \frac{y}{3 x} = 0

(dx)/(dt)=-(11x)/((1000-t))

\frac{d x}{d t} = - \frac{11 x}{( 1000 - t )}