(dy)/(dx)=y^2-16

Lösung

\frac{d y}{d x} = y^{2} - 16

y = \frac{4 e ^{8 c_{1}}}{e ^{8 x} - e ^{8 c_{1}}} + \frac{4 e ^{8 x}}{e ^{8 x} - e ^{8 c_{1}}}

Schritte zur Lösung

\frac{d y}{d x} = y^{2} - 16

Ersetze

\frac{d y}{d x}

mit

y^{'} (x)

y^{^{'}} (x) = y^{2} - 16

Löse nach

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

(\frac{1}{x ^{^{^{'}}}}) = y^{2} - 16

Isoliere

x^{^{'}} : x^{^{'}} = \frac{1}{y ^{2} - 16}

Wenn

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

dann

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

x = \int \frac{1}{y ^{2} - 16} d y

\int \frac{1}{y ^{2} - 16} d y = - \frac{1}{8} (ln (\frac{y}{4} + 1) - ln (\frac{y}{4} - 1)) + c_{1}

x = - \frac{1}{8} (ln (\frac{y}{4} + 1) - ln (\frac{y}{4} - 1)) + c_{1}

Isoliere

y : y = \frac{4 e ^{8 c_{1}}}{e ^{8 x} - e ^{8 c_{1}}} + \frac{4 e ^{8 x}}{e ^{8 x} - e ^{8 c_{1}}}

y = \frac{4 e ^{8 c_{1}}}{e ^{8 x} - e ^{8 c_{1}}} + \frac{4 e ^{8 x}}{e ^{8 x} - e ^{8 c_{1}}}

x \to 0 lim (f (x) - g (x))

\int \frac{1}{x 6 x + 25} d x

\frac{d}{d x} (x - 8 3 x)

\int arctan (8 x) d x

d er i v a t i v e f (x) = x (x^{3} - 8)