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

解答

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

解答

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}}}

求解步骤

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

替代

\frac{d y}{d x}

与

y^{'} (x)

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

求解

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

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

整理

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

若

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

则

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}

整理

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}}}

作图

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