y^'=y^2-3

解答

y^{^{'}} = y^{2} - 3

y = \frac{3 e ^{23 c_{1}}}{e ^{23 t} - e ^{23 c_{1}}} + \frac{3 e ^{23 t}}{e ^{23 t} - e ^{23 c_{1}}}

求解步骤

y^{'} (t) = y^{2} - 3

求解

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

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

整理

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

若

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

则

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

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

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

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

整理

y : y = \frac{3 e ^{23 c_{1}}}{e ^{23 t} - e ^{23 c_{1}}} + \frac{3 e ^{23 t}}{e ^{23 t} - e ^{23 c_{1}}}

y = \frac{3 e ^{23 c_{1}}}{e ^{23 t} - e ^{23 c_{1}}} + \frac{3 e ^{23 t}}{e ^{23 t} - e ^{23 c_{1}}}

a re a y = x^{2} - 4 x, y = 2 x

\int \frac{1}{x 225 - x ^{2}} d x

\int - sin (x) cos^{7} (x) d x

\frac{d y}{d x} = \frac{( x - 4 )}{( y + 3 )}

d er i v a t i v e 5 x (x - 2)