Calculus Examples

$\frac{d P}{d t} = P^{2} - 7 P + 12$ , $P (0) = 1$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{P^{2} - 7 P + 12}$ .

$\frac{1}{P^{2} - 7 P + 12} \frac{d P}{d t} = \frac{1}{P^{2} - 7 P + 12} (P^{2} - 7 P + 12)$

Step 1.2

Cancel the common factor of $P^{2} - 7 P + 12$ .

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Step 1.2.1

Cancel the common factor.

$\frac{1}{P^{2} - 7 P + 12} \frac{d P}{d t} = \frac{1}{P^{2} - 7 P + 12} (P^{2} - 7 P + 12)$

Step 1.2.2

Rewrite the expression.

$\frac{1}{P^{2} - 7 P + 12} \frac{d P}{d t} = 1$

Step 1.3

Rewrite the equation.

$\frac{1}{P^{2} - 7 P + 12} d P = 1 d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{P^{2} - 7 P + 12} d P = \int d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Write the fraction using partial fraction decomposition.

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Step 2.2.1.1

Decompose the fraction and multiply through by the common denominator.

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Step 2.2.1.1.1

Factor $P^{2} - 7 P + 12$ using the AC method.

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Step 2.2.1.1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $12$ and whose sum is $- 7$ .

$- 4, - 3$

Step 2.2.1.1.1.2

Write the factored form using these integers.

$\frac{1}{(P - 4) (P - 3)}$

Step 2.2.1.1.2

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $A$ .

$\frac{A}{P - 4}$

Step 2.2.1.1.3

$\frac{A}{P - 4} + \frac{B}{P - 3}$

Step 2.2.1.1.4

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(P - 4) (P - 3)$ .

$\frac{1 (P - 4) (P - 3)}{(P - 4) (P - 3)} = \frac{(A) (P - 4) (P - 3)}{P - 4} + \frac{(B) (P - 4) (P - 3)}{P - 3}$

Step 2.2.1.1.5

Cancel the common factor of $P - 4$ .

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Step 2.2.1.1.5.1

Cancel the common factor.

$\frac{1 (P - 4) (P - 3)}{(P - 4) (P - 3)} = \frac{(A) (P - 4) (P - 3)}{P - 4} + \frac{(B) (P - 4) (P - 3)}{P - 3}$

Step 2.2.1.1.5.2

Rewrite the expression.

$\frac{1 (P - 3)}{P - 3} = \frac{(A) (P - 4) (P - 3)}{P - 4} + \frac{(B) (P - 4) (P - 3)}{P - 3}$

Step 2.2.1.1.6

Cancel the common factor of $P - 3$ .

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Step 2.2.1.1.6.1

Cancel the common factor.

$\frac{1 (P - 3)}{P - 3} = \frac{(A) (P - 4) (P - 3)}{P - 4} + \frac{(B) (P - 4) (P - 3)}{P - 3}$

Step 2.2.1.1.6.2

Rewrite the expression.

$1 = \frac{(A) (P - 4) (P - 3)}{P - 4} + \frac{(B) (P - 4) (P - 3)}{P - 3}$

Step 2.2.1.1.7

Simplify each term.

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Step 2.2.1.1.7.1

Cancel the common factor of $P - 4$ .

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Step 2.2.1.1.7.1.1

Cancel the common factor.

$1 = \frac{A (P - 4) (P - 3)}{P - 4} + \frac{(B) (P - 4) (P - 3)}{P - 3}$

Step 2.2.1.1.7.1.2

Divide $(A) (P - 3)$ by $1$ .

$1 = (A) (P - 3) + \frac{(B) (P - 4) (P - 3)}{P - 3}$

Step 2.2.1.1.7.2

Apply the distributive property.

$1 = A P + A \cdot - 3 + \frac{(B) (P - 4) (P - 3)}{P - 3}$

Step 2.2.1.1.7.3

Move $- 3$ to the left of $A$ .

$1 = A P - 3 \cdot A + \frac{(B) (P - 4) (P - 3)}{P - 3}$

Step 2.2.1.1.7.4

Cancel the common factor of $P - 3$ .

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Step 2.2.1.1.7.4.1

Cancel the common factor.

$1 = A P - 3 A + \frac{(B) (P - 4) (P - 3)}{P - 3}$

Step 2.2.1.1.7.4.2

Divide $(B) (P - 4)$ by $1$ .

$1 = A P - 3 A + (B) (P - 4)$

Step 2.2.1.1.7.5

Apply the distributive property.

$1 = A P - 3 A + B P + B \cdot - 4$

Step 2.2.1.1.7.6

Move $- 4$ to the left of $B$ .

$1 = A P - 3 A + B P - 4 B$

Step 2.2.1.1.8

Move $- 3 A$ .

$1 = A P + B P - 3 A - 4 B$

Step 2.2.1.2

Create equations for the partial fraction variables and use them to set up a system of equations.

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Step 2.2.1.2.1

Create an equation for the partial fraction variables by equating the coefficients of $P$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = A + B$

Step 2.2.1.2.2

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $P$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = - 3 A - 4 B$

Step 2.2.1.2.3

Set up the system of equations to find the coefficients of the partial fractions.

$0 = A + B$

$1 = - 3 A - 4 B$

$0 = A + B$

$1 = - 3 A - 4 B$

Step 2.2.1.3

Solve the system of equations.

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Step 2.2.1.3.1

Solve for $A$ in $0 = A + B$ .

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Step 2.2.1.3.1.1

Rewrite the equation as $A + B = 0$ .

$A + B = 0$

$1 = - 3 A - 4 B$

Step 2.2.1.3.1.2

Subtract $B$ from both sides of the equation.

$A = - B$

$1 = - 3 A - 4 B$

$A = - B$

$1 = - 3 A - 4 B$

Step 2.2.1.3.2

Replace all occurrences of $A$ with $- B$ in each equation.

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Step 2.2.1.3.2.1

Replace all occurrences of $A$ in $1 = - 3 A - 4 B$ with $- B$ .

$1 = - 3 (- B) - 4 B$

$A = - B$

Step 2.2.1.3.2.2

Simplify the right side.

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Step 2.2.1.3.2.2.1

Simplify $- 3 (- B) - 4 B$ .

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Step 2.2.1.3.2.2.1.1

Multiply $- 1$ by $- 3$ .

$1 = 3 B - 4 B$

$A = - B$

Step 2.2.1.3.2.2.1.2

Subtract $4 B$ from $3 B$ .

$1 = - B$

$A = - B$

$1 = - B$

$A = - B$

$1 = - B$

$A = - B$

$1 = - B$

$A = - B$

Step 2.2.1.3.3

Solve for $B$ in $1 = - B$ .

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Step 2.2.1.3.3.1

Rewrite the equation as $- B = 1$ .

$- B = 1$

$A = - B$

Step 2.2.1.3.3.2

Divide each term in $- B = 1$ by $- 1$ and simplify.

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Step 2.2.1.3.3.2.1

Divide each term in $- B = 1$ by $- 1$ .

$\frac{- B}{- 1} = \frac{1}{- 1}$

$A = - B$

Step 2.2.1.3.3.2.2

Simplify the left side.

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Step 2.2.1.3.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{B}{1} = \frac{1}{- 1}$

$A = - B$

Step 2.2.1.3.3.2.2.2

Divide $B$ by $1$ .

$B = \frac{1}{- 1}$

$A = - B$

$B = \frac{1}{- 1}$

$A = - B$

Step 2.2.1.3.3.2.3

Simplify the right side.

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Step 2.2.1.3.3.2.3.1

Divide $1$ by $- 1$ .

$B = - 1$

$A = - B$

$B = - 1$

$A = - B$

$B = - 1$

$A = - B$

$B = - 1$

$A = - B$

Step 2.2.1.3.4

Replace all occurrences of $B$ with $- 1$ in each equation.

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Step 2.2.1.3.4.1

Replace all occurrences of $B$ in $A = - B$ with $- 1$ .

$A = - (- 1)$

$B = - 1$

Step 2.2.1.3.4.2

Simplify the right side.

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Step 2.2.1.3.4.2.1

Multiply $- 1$ by $- 1$ .

$A = 1$

$B = - 1$

$A = 1$

$B = - 1$

$A = 1$

$B = - 1$

Step 2.2.1.3.5

List all of the solutions.

$A = 1, B = - 1$

Step 2.2.1.4

Replace each of the partial fraction coefficients in $\frac{A}{P - 4} + \frac{B}{P - 3}$ with the values found for $A$ and $B$ .

$\frac{1}{P - 4} + \frac{- 1}{P - 3}$

Step 2.2.1.5

Move the negative in front of the fraction.

$\int \frac{1}{P - 4} - \frac{1}{P - 3} d P = \int d t$

Step 2.2.2

Split the single integral into multiple integrals.

$\int \frac{1}{P - 4} d P + \int - \frac{1}{P - 3} d P = \int d t$

Step 2.2.3

Let $u_{1} = P - 4$ . Then $d u_{1} = d P$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.2.3.1

Let $u_{1} = P - 4$ . Find $\frac{d u_{1}}{d P}$ .

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Step 2.2.3.1.1

Differentiate $P - 4$ .

$\frac{d}{d P} [P - 4]$

Step 2.2.3.1.2

By the Sum Rule, the derivative of $P - 4$ with respect to $P$ is $\frac{d}{d P} [P] + \frac{d}{d P} [- 4]$ .

$\frac{d}{d P} [P] + \frac{d}{d P} [- 4]$

Step 2.2.3.1.3

Differentiate using the Power Rule which states that $\frac{d}{d P} [P^{n}]$ is $n P^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d P} [- 4]$

Step 2.2.3.1.4

Since $- 4$ is constant with respect to $P$ , the derivative of $- 4$ with respect to $P$ is $0$ .

$1 + 0$

Step 2.2.3.1.5

Add $1$ and $0$ .

$1$

Step 2.2.3.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \frac{1}{u_{1}} d u_{1} + \int - \frac{1}{P - 3} d P = \int d t$

Step 2.2.4

The integral of $\frac{1}{u_{1}}$ with respect to $u_{1}$ is $\ln (| u_{1} |)$ .

$\ln (| u_{1} |) + C_{1} + \int - \frac{1}{P - 3} d P = \int d t$

Step 2.2.5

Since $- 1$ is constant with respect to $P$ , move $- 1$ out of the integral.

$\ln (| u_{1} |) + C_{1} - \int \frac{1}{P - 3} d P = \int d t$

Step 2.2.6

Let $u_{2} = P - 3$ . Then $d u_{2} = d P$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.2.6.1

Let $u_{2} = P - 3$ . Find $\frac{d u_{2}}{d P}$ .

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Step 2.2.6.1.1

Differentiate $P - 3$ .

$\frac{d}{d P} [P - 3]$

Step 2.2.6.1.2

By the Sum Rule, the derivative of $P - 3$ with respect to $P$ is $\frac{d}{d P} [P] + \frac{d}{d P} [- 3]$ .

$\frac{d}{d P} [P] + \frac{d}{d P} [- 3]$

Step 2.2.6.1.3

Differentiate using the Power Rule which states that $\frac{d}{d P} [P^{n}]$ is $n P^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d P} [- 3]$

Step 2.2.6.1.4

Since $- 3$ is constant with respect to $P$ , the derivative of $- 3$ with respect to $P$ is $0$ .

$1 + 0$

Step 2.2.6.1.5

Add $1$ and $0$ .

$1$

Step 2.2.6.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\ln (| u_{1} |) + C_{1} - \int \frac{1}{u_{2}} d u_{2} = \int d t$

Step 2.2.7

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$\ln (| u_{1} |) + C_{1} - (\ln (| u_{2} |) + C_{2}) = \int d t$

Step 2.2.8

Simplify.

$\ln (| u_{1} |) - \ln (| u_{2} |) + C_{3} = \int d t$

Step 2.2.9

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{| u_{1} |}{| u_{2} |}) + C_{3} = \int d t$

Step 2.2.10

Substitute back in for each integration substitution variable.

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Step 2.2.10.1

Replace all occurrences of $u_{1}$ with $P - 4$ .

$\ln (\frac{| P - 4 |}{| u_{2} |}) + C_{3} = \int d t$

Step 2.2.10.2

Replace all occurrences of $u_{2}$ with $P - 3$ .

$\ln (\frac{| P - 4 |}{| P - 3 |}) + C_{3} = \int d t$

Step 2.3

Apply the constant rule.

$\ln (\frac{| P - 4 |}{| P - 3 |}) + C_{3} = t + C_{4}$

Step 2.4

Group the constant of integration on the right side as $C$ .

$\ln (\frac{| P - 4 |}{| P - 3 |}) = t + C$

Step 3

Solve for $P$ .

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Step 3.1

To solve for $P$ , rewrite the equation using properties of logarithms.

$e^{\ln (\frac{| P - 4 |}{| P - 3 |})} = e^{t + C}$

Step 3.2

Rewrite $\ln (\frac{| P - 4 |}{| P - 3 |}) = t + C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{t + C} = \frac{| P - 4 |}{| P - 3 |}$

Step 3.3

Solve for $P$ .

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Step 3.3.1

Rewrite the equation as $\frac{| P - 4 |}{| P - 3 |} = e^{t + C}$ .

$\frac{| P - 4 |}{| P - 3 |} = e^{t + C}$

Step 3.3.2

Multiply both sides by $| P - 3 |$ .

$\frac{| P - 4 |}{| P - 3 |} | P - 3 | = e^{t + C} | P - 3 |$

Step 3.3.3

Simplify the left side.

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Step 3.3.3.1

Cancel the common factor of $| P - 3 |$ .

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Step 3.3.3.1.1

Cancel the common factor.

$\frac{| P - 4 |}{| P - 3 |} | P - 3 | = e^{t + C} | P - 3 |$

Step 3.3.3.1.2

Rewrite the expression.

$| P - 4 | = e^{t + C} | P - 3 |$

Step 3.3.4

Solve for $P$ .

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Step 3.3.4.1

Rewrite the equation as $e^{t + C} | P - 3 | = | P - 4 |$ .

$e^{t + C} | P - 3 | = | P - 4 |$

Step 3.3.4.2

Divide each term in $e^{t + C} | P - 3 | = | P - 4 |$ by $e^{t + C}$ and simplify.

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Step 3.3.4.2.1

Divide each term in $e^{t + C} | P - 3 | = | P - 4 |$ by $e^{t + C}$ .

$\frac{e^{t + C} | P - 3 |}{e^{t + C}} = \frac{| P - 4 |}{e^{t + C}}$

Step 3.3.4.2.2

Simplify the left side.

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Step 3.3.4.2.2.1

Cancel the common factor of $e^{t + C}$ .

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Step 3.3.4.2.2.1.1

Cancel the common factor.

$\frac{e^{t + C} | P - 3 |}{e^{t + C}} = \frac{| P - 4 |}{e^{t + C}}$

Step 3.3.4.2.2.1.2

Divide $| P - 3 |$ by $1$ .

$| P - 3 | = \frac{| P - 4 |}{e^{t + C}}$

Step 3.3.4.3

Rewrite the absolute value equation as four equations without absolute value bars.

$P - 3 = \frac{P - 4}{e^{t + C}}$

$P - 3 = - \frac{P - 4}{e^{t + C}}$

$- (P - 3) = \frac{P - 4}{e^{t + C}}$

$- (P - 3) = - \frac{P - 4}{e^{t + C}}$

Step 3.3.4.4

After simplifying, there are only two unique equations to be solved.

$P - 3 = \frac{P - 4}{e^{t + C}}$

$P - 3 = - \frac{P - 4}{e^{t + C}}$

Step 3.3.4.5

Solve $P - 3 = \frac{P - 4}{e^{t + C}}$ for $P$ .

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Step 3.3.4.5.1

Multiply both sides by $e^{t + C}$ .

$(P - 3) e^{t + C} = \frac{P - 4}{e^{t + C}} e^{t + C}$

Step 3.3.4.5.2

Simplify.

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Step 3.3.4.5.2.1

Simplify the left side.

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Step 3.3.4.5.2.1.1

Simplify $(P - 3) e^{t + C}$ .

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Step 3.3.4.5.2.1.1.1

Apply the distributive property.

$P e^{t + C} - 3 e^{t + C} = \frac{P - 4}{e^{t + C}} e^{t + C}$

Step 3.3.4.5.2.1.1.2

Simplify with commuting.

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Step 3.3.4.5.2.1.1.2.1

Reorder $t$ and $C$ .

$P e^{C + t} - 3 e^{t + C} = \frac{P - 4}{e^{t + C}} e^{t + C}$

Step 3.3.4.5.2.1.1.2.2

Reorder $t$ and $C$ .

$P e^{C + t} - 3 e^{C + t} = \frac{P - 4}{e^{t + C}} e^{t + C}$

Step 3.3.4.5.2.2

Simplify the right side.

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Step 3.3.4.5.2.2.1

Cancel the common factor of $e^{t + C}$ .

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Step 3.3.4.5.2.2.1.1

Cancel the common factor.

$P e^{C + t} - 3 e^{C + t} = \frac{P - 4}{e^{t + C}} e^{t + C}$

Step 3.3.4.5.2.2.1.2

Rewrite the expression.

$P e^{C + t} - 3 e^{C + t} = P - 4$

Step 3.3.4.5.3

Solve for $P$ .

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Step 3.3.4.5.3.1

Subtract $P$ from both sides of the equation.

$P e^{C + t} - 3 e^{C + t} - P = - 4$

Step 3.3.4.5.3.2

Add $3 e^{C + t}$ to both sides of the equation.

$P e^{C + t} - P = - 4 + 3 e^{C + t}$

Step 3.3.4.5.3.3

Factor $P$ out of $P e^{C + t} - P$ .

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Step 3.3.4.5.3.3.1

Factor $P$ out of $P e^{C + t}$ .

$P e^{C + t} - P = - 4 + 3 e^{C + t}$

Step 3.3.4.5.3.3.2

Factor $P$ out of $- P$ .

$P e^{C + t} + P \cdot - 1 = - 4 + 3 e^{C + t}$

Step 3.3.4.5.3.3.3

Factor $P$ out of $P e^{C + t} + P \cdot - 1$ .

$P (e^{C + t} - 1) = - 4 + 3 e^{C + t}$

Step 3.3.4.5.3.4

Divide each term in $P (e^{C + t} - 1) = - 4 + 3 e^{C + t}$ by $e^{C + t} - 1$ and simplify.

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Step 3.3.4.5.3.4.1

Divide each term in $P (e^{C + t} - 1) = - 4 + 3 e^{C + t}$ by $e^{C + t} - 1$ .

$\frac{P (e^{C + t} - 1)}{e^{C + t} - 1} = \frac{- 4}{e^{C + t} - 1} + \frac{3 e^{C + t}}{e^{C + t} - 1}$

Step 3.3.4.5.3.4.2

Simplify the left side.

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Step 3.3.4.5.3.4.2.1

Cancel the common factor of $e^{C + t} - 1$ .

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Step 3.3.4.5.3.4.2.1.1

Cancel the common factor.

$\frac{P (e^{C + t} - 1)}{e^{C + t} - 1} = \frac{- 4}{e^{C + t} - 1} + \frac{3 e^{C + t}}{e^{C + t} - 1}$

Step 3.3.4.5.3.4.2.1.2

Divide $P$ by $1$ .

$P = \frac{- 4}{e^{C + t} - 1} + \frac{3 e^{C + t}}{e^{C + t} - 1}$

Step 3.3.4.5.3.4.3

Simplify the right side.

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Step 3.3.4.5.3.4.3.1

Combine the numerators over the common denominator.

$P = \frac{- 4 + 3 e^{C + t}}{e^{C + t} - 1}$

Step 3.3.4.5.3.4.3.2

Rewrite $- 4$ as $- 1 (4)$ .

$P = \frac{- 1 (4) + 3 e^{C + t}}{e^{C + t} - 1}$

Step 3.3.4.5.3.4.3.3

Factor $- 1$ out of $3 e^{C + t}$ .

$P = \frac{- 1 (4) - (- 3 e^{C + t})}{e^{C + t} - 1}$

Step 3.3.4.5.3.4.3.4

Factor $- 1$ out of $- 1 (4) - (- 3 e^{C + t})$ .

$P = \frac{- 1 (4 - 3 e^{C + t})}{e^{C + t} - 1}$

Step 3.3.4.5.3.4.3.5

Move the negative in front of the fraction.

$P = - \frac{4 - 3 e^{C + t}}{e^{C + t} - 1}$

Step 3.3.4.6

Solve $P - 3 = - \frac{P - 4}{e^{t + C}}$ for $P$ .

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Step 3.3.4.6.1

Multiply both sides by $e^{t + C}$ .

$(P - 3) e^{t + C} = - \frac{P - 4}{e^{t + C}} e^{t + C}$

Step 3.3.4.6.2

Simplify.

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Step 3.3.4.6.2.1

Simplify the left side.

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Step 3.3.4.6.2.1.1

Simplify $(P - 3) e^{t + C}$ .

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Step 3.3.4.6.2.1.1.1

Apply the distributive property.

$P e^{t + C} - 3 e^{t + C} = - \frac{P - 4}{e^{t + C}} e^{t + C}$

Step 3.3.4.6.2.1.1.2

Simplify with commuting.

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Step 3.3.4.6.2.1.1.2.1

Reorder $t$ and $C$ .

$P e^{C + t} - 3 e^{t + C} = - \frac{P - 4}{e^{t + C}} e^{t + C}$

Step 3.3.4.6.2.1.1.2.2

Reorder $t$ and $C$ .

$P e^{C + t} - 3 e^{C + t} = - \frac{P - 4}{e^{t + C}} e^{t + C}$

Step 3.3.4.6.2.2

Simplify the right side.

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Step 3.3.4.6.2.2.1

Simplify $- \frac{P - 4}{e^{t + C}} e^{t + C}$ .

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Step 3.3.4.6.2.2.1.1

Cancel the common factor of $e^{t + C}$ .

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Step 3.3.4.6.2.2.1.1.1

Move the leading negative in $- \frac{P - 4}{e^{t + C}}$ into the numerator.

$P e^{C + t} - 3 e^{C + t} = \frac{- (P - 4)}{e^{t + C}} e^{t + C}$

Step 3.3.4.6.2.2.1.1.2

Cancel the common factor.

$P e^{C + t} - 3 e^{C + t} = \frac{- (P - 4)}{e^{t + C}} e^{t + C}$

Step 3.3.4.6.2.2.1.1.3

Rewrite the expression.

$P e^{C + t} - 3 e^{C + t} = - (P - 4)$

Step 3.3.4.6.2.2.1.2

Apply the distributive property.

$P e^{C + t} - 3 e^{C + t} = - P - - 4$

Step 3.3.4.6.2.2.1.3

Multiply $- 1$ by $- 4$ .

$P e^{C + t} - 3 e^{C + t} = - P + 4$

Step 3.3.4.6.3

Solve for $P$ .

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Step 3.3.4.6.3.1

Add $P$ to both sides of the equation.

$P e^{C + t} - 3 e^{C + t} + P = 4$

Step 3.3.4.6.3.2

Add $3 e^{C + t}$ to both sides of the equation.

$P e^{C + t} + P = 4 + 3 e^{C + t}$

Step 3.3.4.6.3.3

Factor $P$ out of $P e^{C + t} + P$ .

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Step 3.3.4.6.3.3.1

Factor $P$ out of $P e^{C + t}$ .

$P e^{C + t} + P = 4 + 3 e^{C + t}$

Step 3.3.4.6.3.3.2

Raise $P$ to the power of $1$ .

$P e^{C + t} + P = 4 + 3 e^{C + t}$

Step 3.3.4.6.3.3.3

Factor $P$ out of $P^{1}$ .

$P e^{C + t} + P \cdot 1 = 4 + 3 e^{C + t}$

Step 3.3.4.6.3.3.4

Factor $P$ out of $P e^{C + t} + P \cdot 1$ .

$P (e^{C + t} + 1) = 4 + 3 e^{C + t}$

Step 3.3.4.6.3.4

Divide each term in $P (e^{C + t} + 1) = 4 + 3 e^{C + t}$ by $e^{C + t} + 1$ and simplify.

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Step 3.3.4.6.3.4.1

Divide each term in $P (e^{C + t} + 1) = 4 + 3 e^{C + t}$ by $e^{C + t} + 1$ .

$\frac{P (e^{C + t} + 1)}{e^{C + t} + 1} = \frac{4}{e^{C + t} + 1} + \frac{3 e^{C + t}}{e^{C + t} + 1}$

Step 3.3.4.6.3.4.2

Simplify the left side.

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Step 3.3.4.6.3.4.2.1

Cancel the common factor of $e^{C + t} + 1$ .

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Step 3.3.4.6.3.4.2.1.1

Cancel the common factor.

$\frac{P (e^{C + t} + 1)}{e^{C + t} + 1} = \frac{4}{e^{C + t} + 1} + \frac{3 e^{C + t}}{e^{C + t} + 1}$

Step 3.3.4.6.3.4.2.1.2

Divide $P$ by $1$ .

$P = \frac{4}{e^{C + t} + 1} + \frac{3 e^{C + t}}{e^{C + t} + 1}$

Step 3.3.4.6.3.4.3

Simplify the right side.

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Step 3.3.4.6.3.4.3.1

Combine the numerators over the common denominator.

$P = \frac{4 + 3 e^{C + t}}{e^{C + t} + 1}$

Step 3.3.4.7

List all of the solutions.

$P = - \frac{4 - 3 e^{C + t}}{e^{C + t} - 1}$

$P = \frac{4 + 3 e^{C + t}}{e^{C + t} + 1}$

$P = - \frac{4 - 3 e^{C + t}}{e^{C + t} - 1}$

$P = \frac{4 + 3 e^{C + t}}{e^{C + t} + 1}$

$P = - \frac{4 - 3 e^{C + t}}{e^{C + t} - 1}$

$P = \frac{4 + 3 e^{C + t}}{e^{C + t} + 1}$

$P = - \frac{4 - 3 e^{C + t}}{e^{C + t} - 1}$

$P = \frac{4 + 3 e^{C + t}}{e^{C + t} + 1}$

Step 4

Group the constant terms together.

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Step 4.1

Reorder terms.

$P = - \frac{4 - 3 e^{t + C}}{e^{C + t} - 1}$

$P = \frac{4 + 3 e^{C + t}}{e^{C + t} + 1}$

Step 4.2

Rewrite $e^{t + C}$ as $e^{t} e^{C}$ .

$P = - \frac{4 - 3 (e^{t} e^{C})}{e^{C + t} - 1}$

$P = \frac{4 + 3 e^{C + t}}{e^{C + t} + 1}$

Step 4.3

Reorder $e^{t}$ and $e^{C}$ .

$P = - \frac{4 - 3 (e^{C} e^{t})}{e^{C + t} - 1}$

$P = \frac{4 + 3 e^{C + t}}{e^{C + t} + 1}$

Step 4.4

Reorder terms.

$P = - \frac{4 - 3 (e^{C} e^{t})}{e^{t + C} - 1}$

$P = \frac{4 + 3 e^{C + t}}{e^{C + t} + 1}$

Step 4.5

Rewrite $e^{t + C}$ as $e^{t} e^{C}$ .

$P = - \frac{4 - 3 (e^{C} e^{t})}{e^{t} e^{C} - 1}$

$P = \frac{4 + 3 e^{C + t}}{e^{C + t} + 1}$

Step 4.6

Reorder $e^{t}$ and $e^{C}$ .

$P = - \frac{4 - 3 (e^{C} e^{t})}{e^{C} e^{t} - 1}$

$P = \frac{4 + 3 e^{C + t}}{e^{C + t} + 1}$

Step 4.7

Reorder terms.

$P = - \frac{4 - 3 (e^{C} e^{t})}{e^{C} e^{t} - 1}$

$P = \frac{4 + 3 e^{t + C}}{e^{C + t} + 1}$

Step 4.8

Rewrite $e^{t + C}$ as $e^{t} e^{C}$ .

$P = - \frac{4 - 3 (e^{C} e^{t})}{e^{C} e^{t} - 1}$

$P = \frac{4 + 3 (e^{t} e^{C})}{e^{C + t} + 1}$

Step 4.9

Reorder $e^{t}$ and $e^{C}$ .

$P = - \frac{4 - 3 (e^{C} e^{t})}{e^{C} e^{t} - 1}$

$P = \frac{4 + 3 (e^{C} e^{t})}{e^{C + t} + 1}$

Step 4.10

Reorder terms.

$P = - \frac{4 - 3 (e^{C} e^{t})}{e^{C} e^{t} - 1}$

$P = \frac{4 + 3 (e^{C} e^{t})}{e^{t + C} + 1}$

Step 4.11

Rewrite $e^{t + C}$ as $e^{t} e^{C}$ .

$P = - \frac{4 - 3 (e^{C} e^{t})}{e^{C} e^{t} - 1}$

$P = \frac{4 + 3 (e^{C} e^{t})}{e^{t} e^{C} + 1}$

Step 4.12

Reorder $e^{t}$ and $e^{C}$ .

$P = - \frac{4 - 3 (e^{C} e^{t})}{e^{C} e^{t} - 1}$

$P = \frac{4 + 3 (e^{C} e^{t})}{e^{C} e^{t} + 1}$

$P = - \frac{4 - 3 (e^{C} e^{t})}{e^{C} e^{t} - 1}$

$P = \frac{4 + 3 (e^{C} e^{t})}{e^{C} e^{t} + 1}$

Step 5

Since $P$ is positive in the initial condition $(0, 1)$ , only consider $P = - \frac{4 - 3 (e^{C} e^{t})}{e^{C} e^{t} - 1}$ to find the $C$ . Substitute $0$ for $t$ and $1$ for $P$ .

$1 = - \frac{4 - 3 (e^{C} e^{0})}{e^{C} e^{0} - 1}$

Step 6

Solve for $C$ .

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Step 6.1

Rewrite the equation as $- \frac{4 - 3 e^{C} e^{0}}{e^{C} e^{0} - 1} = 1$ .

$- \frac{4 - 3 e^{C} e^{0}}{e^{C} e^{0} - 1} = 1$

Step 6.2

Multiply both sides by $e^{C} e^{0} - 1$ .

$- \frac{4 - 3 e^{C} e^{0}}{e^{C} e^{0} - 1} (e^{C} e^{0} - 1) = 1 (e^{C} e^{0} - 1)$

Step 6.3

Simplify.

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Step 6.3.1

Simplify the left side.

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Step 6.3.1.1

Simplify $- \frac{4 - 3 e^{C} e^{0}}{e^{C} e^{0} - 1} (e^{C} e^{0} - 1)$ .

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Step 6.3.1.1.1

Cancel the common factor of $e^{C} e^{0} - 1$ .

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Step 6.3.1.1.1.1

Move the leading negative in $- \frac{4 - 3 e^{C} e^{0}}{e^{C} e^{0} - 1}$ into the numerator.

$\frac{- (4 - 3 e^{C} e^{0})}{e^{C} e^{0} - 1} (e^{C} e^{0} - 1) = 1 (e^{C} e^{0} - 1)$

Step 6.3.1.1.1.2

Cancel the common factor.

$\frac{- (4 - 3 e^{C} e^{0})}{e^{C} e^{0} - 1} (e^{C} e^{0} - 1) = 1 (e^{C} e^{0} - 1)$

Step 6.3.1.1.1.3

Rewrite the expression.

$- (4 - 3 e^{C} e^{0}) = 1 (e^{C} e^{0} - 1)$

Step 6.3.1.1.2

Multiply $e^{C}$ by $e^{0}$ by adding the exponents.

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Step 6.3.1.1.2.1

Move $e^{0}$ .

$- (4 - 3 (e^{0} e^{C})) = 1 (e^{C} e^{0} - 1)$

Step 6.3.1.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- (4 - 3 e^{0 + C}) = 1 (e^{C} e^{0} - 1)$

Step 6.3.1.1.2.3

Add $0$ and $C$ .

$- (4 - 3 e^{C}) = 1 (e^{C} e^{0} - 1)$

Step 6.3.1.1.3

Apply the distributive property.

$- 1 \cdot 4 - (- 3 e^{C}) = 1 (e^{C} e^{0} - 1)$

Step 6.3.1.1.4

Multiply.

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Step 6.3.1.1.4.1

Multiply $- 1$ by $4$ .

$- 4 - (- 3 e^{C}) = 1 (e^{C} e^{0} - 1)$

Step 6.3.1.1.4.2

Multiply $- 3$ by $- 1$ .

$- 4 + 3 e^{C} = 1 (e^{C} e^{0} - 1)$

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

Simplify $1 (e^{C} e^{0} - 1)$ .

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Step 6.3.2.1.1

Multiply $e^{C} e^{0} - 1$ by $1$ .

$- 4 + 3 e^{C} = e^{C} e^{0} - 1$

Step 6.3.2.1.2

Multiply $e^{C}$ by $e^{0}$ by adding the exponents.

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Step 6.3.2.1.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 4 + 3 e^{C} = e^{C + 0} - 1$

Step 6.3.2.1.2.2

Add $C$ and $0$ .

$- 4 + 3 e^{C} = e^{C} - 1$

Step 6.4

Solve for $C$ .

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Step 6.4.1

Move all terms containing $C$ to the left side of the equation.

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Step 6.4.1.1

Subtract $e^{C}$ from both sides of the equation.

$- 4 + 3 e^{C} - e^{C} = - 1$

Step 6.4.1.2

Subtract $e^{C}$ from $3 e^{C}$ .

$- 4 + 2 e^{C} = - 1$

Step 6.4.2

Move all terms not containing $C$ to the right side of the equation.

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Step 6.4.2.1

Add $4$ to both sides of the equation.

$2 e^{C} = - 1 + 4$

Step 6.4.2.2

Add $- 1$ and $4$ .

$2 e^{C} = 3$

Step 6.4.3

Divide each term in $2 e^{C} = 3$ by $2$ and simplify.

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Step 6.4.3.1

Divide each term in $2 e^{C} = 3$ by $2$ .

$\frac{2 e^{C}}{2} = \frac{3}{2}$

Step 6.4.3.2

Simplify the left side.

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Step 6.4.3.2.1

Cancel the common factor of $2$ .

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Step 6.4.3.2.1.1

Cancel the common factor.

$\frac{2 e^{C}}{2} = \frac{3}{2}$

Step 6.4.3.2.1.2

Divide $e^{C}$ by $1$ .

$e^{C} = \frac{3}{2}$

Step 6.4.4

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{C}) = \ln (\frac{3}{2})$

Step 6.4.5

Expand the left side.

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Step 6.4.5.1

Expand $\ln (e^{C})$ by moving $C$ outside the logarithm.

$C \ln (e) = \ln (\frac{3}{2})$

Step 6.4.5.2

The natural logarithm of $e$ is $1$ .

$C \cdot 1 = \ln (\frac{3}{2})$

Step 6.4.5.3

Multiply $C$ by $1$ .

$C = \ln (\frac{3}{2})$

Step 7

Substitute $\ln (\frac{3}{2})$ for $C$ in $P = - \frac{4 - 3 (e^{C} e^{t})}{e^{C} e^{t} - 1}$ and simplify.

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Step 7.1

Substitute $\ln (\frac{3}{2})$ for $C$ .

$P = - \frac{4 - 3 (e^{\ln (\frac{3}{2})} e^{t})}{e^{\ln (\frac{3}{2})} e^{t} - 1}$

Step 7.2

Multiply $e^{\ln (\frac{3}{2})}$ by $e^{t}$ by adding the exponents.

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Step 7.2.1

Move $e^{t}$ .

$P = - \frac{4 - 3 (e^{t} e^{\ln (\frac{3}{2})})}{e^{\ln (\frac{3}{2})} e^{t} - 1}$

Step 7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$P = - \frac{4 - 3 e^{t + \ln (\frac{3}{2})}}{e^{\ln (\frac{3}{2})} e^{t} - 1}$

Step 7.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$P = - \frac{4 - 3 e^{t + \ln (\frac{3}{2})}}{e^{\ln (\frac{3}{2}) + t} - 1}$