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Popular Calculus >

x^'=1-3x+x^2

Solution

x^{^{'}} = 1 - 3 x + x^{2}

Solution

x = - \frac{3 e ^{5 c_{1}}}{2 ( e ^{5 t} - e ^{5 c_{1}} )} + \frac{5 e ^{5 c_{1}}}{2 ( e ^{5 t} - e ^{5 c_{1}} )} + \frac{3 e ^{5 t}}{2 ( e ^{5 t} - e ^{5 c_{1}} )} + \frac{5 e ^{5 t}}{2 ( e ^{5 t} - e ^{5 c_{1}} )}

Solution steps

x^{'} (t) = 1 - 3 x + x^{2}

Solve for

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

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

Isolate

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

If

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

then

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

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

\int \frac{1}{1 - 3 x + x ^{2}} d x = - \frac{1}{5} (ln (\frac{2 x - 3}{5} + 1) - ln (\frac{2 x - 3}{5} - 1)) + c_{1}

t = - \frac{1}{5} (ln (\frac{2 x - 3}{5} + 1) - ln (\frac{2 x - 3}{5} - 1)) + c_{1}

Isolate

x : x = - \frac{3 e ^{5 c_{1}}}{2 ( e ^{5 t} - e ^{5 c_{1}} )} + \frac{5 e ^{5 c_{1}}}{2 ( e ^{5 t} - e ^{5 c_{1}} )} + \frac{3 e ^{5 t}}{2 ( e ^{5 t} - e ^{5 c_{1}} )} + \frac{5 e ^{5 t}}{2 ( e ^{5 t} - e ^{5 c_{1}} )}

x = - \frac{3 e ^{5 c_{1}}}{2 ( e ^{5 t} - e ^{5 c_{1}} )} + \frac{5 e ^{5 c_{1}}}{2 ( e ^{5 t} - e ^{5 c_{1}} )} + \frac{3 e ^{5 t}}{2 ( e ^{5 t} - e ^{5 c_{1}} )} + \frac{5 e ^{5 t}}{2 ( e ^{5 t} - e ^{5 c_{1}} )}

Graph

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Frequently Asked Questions (FAQ)

What is the solution for x^'=1-3x+x^2 ?
The solution for x^'=1-3x+x^2 is