(dy)/(dx)=y^2-9

Lösung

\frac{d y}{d x} = y^{2} - 9

y = \frac{3 e ^{6 c_{1}}}{e ^{6 x} - e ^{6 c_{1}}} + \frac{3 e ^{6 x}}{e ^{6 x} - e ^{6 c_{1}}}

Schritte zur Lösung

\frac{d y}{d x} = y^{2} - 9

Ersetze

\frac{d y}{d x}

mit

y^{'} (x)

y^{^{'}} (x) = y^{2} - 9

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} - 9

Isoliere

x^{^{'}} : x^{^{'}} = \frac{1}{y ^{2} - 9}

Wenn

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

dann

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

x = \int \frac{1}{y ^{2} - 9} d y

\int \frac{1}{y ^{2} - 9} d y = - \frac{1}{6} (ln (\frac{y}{3} + 1) - ln (\frac{y}{3} - 1)) + c_{1}

x = - \frac{1}{6} (ln (\frac{y}{3} + 1) - ln (\frac{y}{3} - 1)) + c_{1}

Isoliere

y : y = \frac{3 e ^{6 c_{1}}}{e ^{6 x} - e ^{6 c_{1}}} + \frac{3 e ^{6 x}}{e ^{6 x} - e ^{6 c_{1}}}

y = \frac{3 e ^{6 c_{1}}}{e ^{6 x} - e ^{6 c_{1}}} + \frac{3 e ^{6 x}}{e ^{6 x} - e ^{6 c_{1}}}

\frac{d}{d x} (5 x^{3} + x^{2} + \frac{1}{x})

\frac{d}{d x} (2 (5 - 9 x)^{5})

\int \frac{1 - x}{1 - x ^{2}} d x

\int_{0}^{2} (x^{2} - 2 x) d x

\int sin^{2} (x + 4) d x