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人気のある微分積分 >

y^'=y^2-4

解

y^{^{'}} = y^{2} - 4

解

y = \frac{2 e ^{4 c_{1}}}{e ^{4 t} - e ^{4 c_{1}}} + \frac{2 e ^{4 t}}{e ^{4 t} - e ^{4 c_{1}}}

解答ステップ

y^{'} (t) = y^{2} - 4

t

を解く

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

(\frac{1}{t ^{^{^{'}}}}) = y^{2} - 4

分離する

t^{^{'}} : t^{^{'}} = \frac{1}{y ^{2} - 4}

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

であれば

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

t = \int \frac{1}{y ^{2} - 4} d y

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

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

分離する

y : y = \frac{2 e ^{4 c_{1}}}{e ^{4 t} - e ^{4 c_{1}}} + \frac{2 e ^{4 t}}{e ^{4 t} - e ^{4 c_{1}}}

y = \frac{2 e ^{4 c_{1}}}{e ^{4 t} - e ^{4 c_{1}}} + \frac{2 e ^{4 t}}{e ^{4 t} - e ^{4 c_{1}}}

グラフ

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