Solve dp/dt = p-p^2

Question

Solve the equation

Solve for p

Solve for t

p = 0 p = \frac{- 1 + t}{t}

Evaluate

d \times \frac{p}{d t} = p - p^{2}

Multiply the terms

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Multiply the terms

d \times \frac{p}{d t}

Cancel out the common factor d

1 \times \frac{p}{t}

Multiply the terms

\frac{p}{t}

\frac{p}{t} = p - p^{2}

Multiply both sides of the equation by LCD

\frac{p}{t} \times t = (p - p^{2}) t

Simplify the equation

p = (p - p^{2}) t

Simplify the equation

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Evaluate

(p - p^{2}) t

Apply the distributive property

pt - p^{2} t

Multiply the terms

tp - p^{2} t

Multiply the terms

tp - t p^{2}

p = tp - t p^{2}

Move the expression to the left side

p - (tp - t p^{2}) = 0

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

p - tp + t p^{2} = 0

Factor the expression

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Evaluate

p - tp + t p^{2}

Rewrite the expression

p - pt + ptp

Factor out p from the expression

p (1 - t + tp)

p (1 - t + tp) = 0

When the product of factors equals 0,at least one factor is 0

p = 0 1 - t + tp = 0

Solution

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Evaluate

1 - t + tp = 0

Move the expression to the right-hand side and change its sign

tp = 0 - (1 - t)

Subtract the terms

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Evaluate

0 - (1 - t)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

0 - 1 + t

Removing 0 doesn't change the value,so remove it from the expression

- 1 + t

tp = - 1 + t

Divide both sides

\frac{tp}{t} = \frac{- 1 + t}{t}

Divide the numbers

p = \frac{- 1 + t}{t}

p = 0 p = \frac{- 1 + t}{t}

Show Solution