x^'=1-3x+x^2

Lösung

x^{^{'}} = 1 - 3 x + x^{2}

x = - \frac{3 e ^{5 c_{1}}}{2 ( e ^{5 t} - e ^{5 c_{1}} )} + \frac{5 e ^{5 c_{1}}}{2 ( e ^{5 t} - e ^{5 c_{1}} )} + \frac{3 e ^{5 t}}{2 ( e ^{5 t} - e ^{5 c_{1}} )} + \frac{5 e ^{5 t}}{2 ( e ^{5 t} - e ^{5 c_{1}} )}

Schritte zur Lösung

x^{'} (t) = 1 - 3 x + x^{2}

Löse nach

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

(\frac{1}{t ^{^{^{'}}}}) = 1 - 3 x + x^{2}

Isoliere

t^{^{'}} : t^{^{'}} = \frac{1}{1 - 3 x + x ^{2}}

Wenn

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

dann

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

t = \int \frac{1}{1 - 3 x + x ^{2}} d x

\int \frac{1}{1 - 3 x + x ^{2}} d x = - \frac{1}{5} (ln (\frac{2 x - 3}{5} + 1) - ln (\frac{2 x - 3}{5} - 1)) + c_{1}

t = - \frac{1}{5} (ln (\frac{2 x - 3}{5} + 1) - ln (\frac{2 x - 3}{5} - 1)) + c_{1}

Isoliere

x : x = - \frac{3 e ^{5 c_{1}}}{2 ( e ^{5 t} - e ^{5 c_{1}} )} + \frac{5 e ^{5 c_{1}}}{2 ( e ^{5 t} - e ^{5 c_{1}} )} + \frac{3 e ^{5 t}}{2 ( e ^{5 t} - e ^{5 c_{1}} )} + \frac{5 e ^{5 t}}{2 ( e ^{5 t} - e ^{5 c_{1}} )}

x = - \frac{3 e ^{5 c_{1}}}{2 ( e ^{5 t} - e ^{5 c_{1}} )} + \frac{5 e ^{5 c_{1}}}{2 ( e ^{5 t} - e ^{5 c_{1}} )} + \frac{3 e ^{5 t}}{2 ( e ^{5 t} - e ^{5 c_{1}} )} + \frac{5 e ^{5 t}}{2 ( e ^{5 t} - e ^{5 c_{1}} )}