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Popular >

(dy)/(dt)=y^2-2y-2

Solution

\frac{d y}{d t} = y^{2} - 2 y - 2

Solution

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

Solution steps

\frac{d y}{d t} = y^{2} - 2 y - 2

Substitute

\frac{d y}{d t}

with

y^{'} (t)

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

Solve for

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

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

Isolate

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

If

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

then

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

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

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

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

Isolate

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

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

Graph

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Frequently Asked Questions (FAQ)

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