Calculus Examples

$\frac{d y}{d x} = 0.05 y (500 - y)$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{y (500 - y)}$ .

$\frac{1}{y (500 - y)} \frac{d y}{d x} = \frac{1}{y (500 - y)} (0.05 y (500 - y))$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{y (500 - y)} \frac{d y}{d x} = 0.05 \frac{1}{y (500 - y)} (y (500 - y))$

Step 1.2.2

Combine $0.05$ and $\frac{1}{y (500 - y)}$ .

$\frac{1}{y (500 - y)} \frac{d y}{d x} = \frac{0.05}{y (500 - y)} (y (500 - y))$

Step 1.2.3

Cancel the common factor of $0.05$ .

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Step 1.2.3.1

Cancel the common factor.

$\frac{1}{y (500 - y)} \frac{d y}{d x} = \frac{0.05}{y (500 - y)} (y (500 - y))$

Step 1.2.3.2

Rewrite the expression.

$\frac{1}{y (500 - y)} \frac{d y}{d x} = \frac{1}{y (500 - y)} \cdot \frac{y (500 - y)}{20}$

Step 1.2.4

Cancel the common factor of $y (500 - y)$ .

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Step 1.2.4.1

Cancel the common factor.

$\frac{1}{y (500 - y)} \frac{d y}{d x} = \frac{1}{y (500 - y)} \cdot \frac{y (500 - y)}{20}$

Step 1.2.4.2

Rewrite the expression.

$\frac{1}{y (500 - y)} \frac{d y}{d x} = \frac{1}{20}$

Step 1.3

Rewrite the equation.

$\frac{1}{y (500 - y)} d y = \frac{1}{20} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y (500 - y)} d y = \int \frac{1}{20} d x$

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

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 $B$ .

$\frac{A}{y} + \frac{B}{500 - y}$

Step 2.2.1.1.2

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $y (500 - y)$ .

$\frac{1 (y (500 - y))}{y (500 - y)} = \frac{A (y (500 - y))}{y} + \frac{(B) (y (500 - y))}{500 - y}$

Step 2.2.1.1.3

Cancel the common factor of $y$ .

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Step 2.2.1.1.3.1

Cancel the common factor.

$\frac{1 (y (500 - y))}{y (500 - y)} = \frac{A (y (500 - y))}{y} + \frac{(B) (y (500 - y))}{500 - y}$

Step 2.2.1.1.3.2

Rewrite the expression.

$\frac{1 (500 - y)}{500 - y} = \frac{A (y (500 - y))}{y} + \frac{(B) (y (500 - y))}{500 - y}$

Step 2.2.1.1.4

Cancel the common factor of $500 - y$ .

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Step 2.2.1.1.4.1

Cancel the common factor.

$\frac{1 (500 - y)}{500 - y} = \frac{A (y (500 - y))}{y} + \frac{(B) (y (500 - y))}{500 - y}$

Step 2.2.1.1.4.2

Rewrite the expression.

$1 = \frac{A (y (500 - y))}{y} + \frac{(B) (y (500 - y))}{500 - y}$

Step 2.2.1.1.5

Simplify each term.

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Step 2.2.1.1.5.1

Cancel the common factor of $y$ .

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Step 2.2.1.1.5.1.1

Cancel the common factor.

$1 = \frac{A (y (500 - y))}{y} + \frac{(B) (y (500 - y))}{500 - y}$

Step 2.2.1.1.5.1.2

Divide $A (500 - y)$ by $1$ .

$1 = A (500 - y) + \frac{(B) (y (500 - y))}{500 - y}$

Step 2.2.1.1.5.2

Apply the distributive property.

$1 = A \cdot 500 + A (- y) + \frac{(B) (y (500 - y))}{500 - y}$

Step 2.2.1.1.5.3

Move $500$ to the left of $A$ .

$1 = 500 \cdot A + A (- y) + \frac{(B) (y (500 - y))}{500 - y}$

Step 2.2.1.1.5.4

Rewrite using the commutative property of multiplication.

$1 = 500 \cdot A - A y + \frac{(B) (y (500 - y))}{500 - y}$

Step 2.2.1.1.5.5

Cancel the common factor of $500 - y$ .

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Step 2.2.1.1.5.5.1

Cancel the common factor.

$1 = 500 A - A y + \frac{B (y (500 - y))}{500 - y}$

Step 2.2.1.1.5.5.2

Divide $(B) (y)$ by $1$ .

$1 = 500 A - A y + (B) (y)$

$1 = 500 A - A y + B y$

Step 2.2.1.1.6

Simplify the expression.

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Step 2.2.1.1.6.1

Move $A$ .

$1 = 500 A - y A + B y$

Step 2.2.1.1.6.2

Reorder $B$ and $y$ .

$1 = 500 A - y A + B y$

Step 2.2.1.1.6.3

Move $500 A$ .

$1 = - y A + B y + 500 A$

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 $y$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = - 1 A + B$

Step 2.2.1.2.2

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

$1 = 500 A$

Step 2.2.1.2.3

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

$0 = - 1 A + B$

$1 = 500 A$

$0 = - 1 A + B$

$1 = 500 A$

Step 2.2.1.3

Solve the system of equations.

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Step 2.2.1.3.1

Solve for $A$ in $1 = 500 A$ .

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Step 2.2.1.3.1.1

Rewrite the equation as $500 A = 1$ .

$500 A = 1$

$0 = - 1 A + B$

Step 2.2.1.3.1.2

Divide each term in $500 A = 1$ by $500$ and simplify.

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Step 2.2.1.3.1.2.1

Divide each term in $500 A = 1$ by $500$ .

$\frac{500 A}{500} = \frac{1}{500}$

$0 = - 1 A + B$

Step 2.2.1.3.1.2.2

Simplify the left side.

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Step 2.2.1.3.1.2.2.1

Cancel the common factor of $500$ .

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Step 2.2.1.3.1.2.2.1.1

Cancel the common factor.

$\frac{500 A}{500} = \frac{1}{500}$

$0 = - 1 A + B$

Step 2.2.1.3.1.2.2.1.2

Divide $A$ by $1$ .

$A = \frac{1}{500}$

$0 = - 1 A + B$

$A = \frac{1}{500}$

$0 = - 1 A + B$

$A = \frac{1}{500}$

$0 = - 1 A + B$

$A = \frac{1}{500}$

$0 = - 1 A + B$

$A = \frac{1}{500}$

$0 = - 1 A + B$

Step 2.2.1.3.2

Replace all occurrences of $A$ with $\frac{1}{500}$ in each equation.

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Step 2.2.1.3.2.1

Replace all occurrences of $A$ in $0 = - 1 A + B$ with $\frac{1}{500}$ .

$0 = - 1 (\frac{1}{500}) + B$

$A = \frac{1}{500}$

Step 2.2.1.3.2.2

Simplify the right side.

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Step 2.2.1.3.2.2.1

Rewrite $- 1 (\frac{1}{500})$ as $- (\frac{1}{500})$ .

$0 = - \frac{1}{500} + B$

$A = \frac{1}{500}$

$0 = - \frac{1}{500} + B$

$A = \frac{1}{500}$

$0 = - \frac{1}{500} + B$

$A = \frac{1}{500}$

Step 2.2.1.3.3

Solve for $B$ in $0 = - \frac{1}{500} + B$ .

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Step 2.2.1.3.3.1

Rewrite the equation as $- \frac{1}{500} + B = 0$ .

$- \frac{1}{500} + B = 0$

$A = \frac{1}{500}$

Step 2.2.1.3.3.2

Add $\frac{1}{500}$ to both sides of the equation.

$B = \frac{1}{500}$

$A = \frac{1}{500}$

$B = \frac{1}{500}$

$A = \frac{1}{500}$

Step 2.2.1.3.4

Solve the system of equations.

$B = \frac{1}{500} A = \frac{1}{500}$

Step 2.2.1.3.5

List all of the solutions.

$B = \frac{1}{500}, A = \frac{1}{500}$

Step 2.2.1.4

Replace each of the partial fraction coefficients in $\frac{A}{y} + \frac{B}{500 - y}$ with the values found for $A$ and $B$ .

$\frac{\frac{1}{500}}{y} + \frac{\frac{1}{500}}{500 - y}$

Step 2.2.1.5

Simplify.

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Step 2.2.1.5.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{500} \cdot \frac{1}{y} + \frac{\frac{1}{500}}{500 - y}$

Step 2.2.1.5.2

Multiply $\frac{1}{500}$ by $\frac{1}{y}$ .

$\frac{1}{500 y} + \frac{\frac{1}{500}}{500 - y}$

Step 2.2.1.5.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{500 y} + \frac{1}{500} \cdot \frac{1}{500 - y}$

Step 2.2.1.5.4

Multiply $\frac{1}{500}$ by $\frac{1}{500 - y}$ .

$\int \frac{1}{500 y} + \frac{1}{500 (500 - y)} d y = \int \frac{1}{20} d x$

Step 2.2.2

Split the single integral into multiple integrals.

$\int \frac{1}{500 y} d y + \int \frac{1}{500 (500 - y)} d y = \int \frac{1}{20} d x$

Step 2.2.3

Since $\frac{1}{500}$ is constant with respect to $y$ , move $\frac{1}{500}$ out of the integral.

$\frac{1}{500} \int \frac{1}{y} d y + \int \frac{1}{500 (500 - y)} d y = \int \frac{1}{20} d x$

Step 2.2.4

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

$\frac{1}{500} (\ln (| y |) + C_{1}) + \int \frac{1}{500 (500 - y)} d y = \int \frac{1}{20} d x$

Step 2.2.5

Since $\frac{1}{500}$ is constant with respect to $y$ , move $\frac{1}{500}$ out of the integral.

$\frac{1}{500} (\ln (| y |) + C_{1}) + \frac{1}{500} \int \frac{1}{500 - y} d y = \int \frac{1}{20} d x$

Step 2.2.6

Let $u = 500 - y$ . Then $d u = - d y$ , so $- d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.6.1

Let $u = 500 - y$ . Find $\frac{d u}{d y}$ .

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Step 2.2.6.1.1

Rewrite.

$\frac{- 1}{1}$

Step 2.2.6.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 2.2.6.2

Rewrite the problem using $u$ and $d u$ .

$\frac{1}{500} (\ln (| y |) + C_{1}) + \frac{1}{500} \int \frac{- 1}{u} d u = \int \frac{1}{20} d x$

Step 2.2.7

Move the negative in front of the fraction.

$\frac{1}{500} (\ln (| y |) + C_{1}) + \frac{1}{500} \int - \frac{1}{u} d u = \int \frac{1}{20} d x$

Step 2.2.8

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

$\frac{1}{500} (\ln (| y |) + C_{1}) + \frac{1}{500} (- \int \frac{1}{u} d u) = \int \frac{1}{20} d x$

Step 2.2.9

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

$\frac{1}{500} (\ln (| y |) + C_{1}) + \frac{1}{500} (- (\ln (| u |) + C_{2})) = \int \frac{1}{20} d x$

Step 2.2.10

Simplify.

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Step 2.2.10.1

Simplify.

$\frac{1}{500} \ln (| y |) + \frac{1}{500} (- \ln (| u |)) + C_{3} = \int \frac{1}{20} d x$

Step 2.2.10.2

Combine $\frac{1}{500}$ and $\ln (| u |)$ .

$\frac{1}{500} \ln (| y |) - \frac{\ln (| u |)}{500} + C_{3} = \int \frac{1}{20} d x$

Step 2.2.11

Replace all occurrences of $u$ with $500 - y$ .

$\frac{1}{500} \ln (| y |) - \frac{\ln (| 500 - y |)}{500} + C_{3} = \int \frac{1}{20} d x$

Step 2.2.12

Reorder terms.

$\frac{1}{500} \ln (| y |) - \frac{1}{500} \ln (| 500 - y |) + C_{3} = \int \frac{1}{20} d x$

Step 2.3

Apply the constant rule.

$\frac{1}{500} \ln (| y |) - \frac{1}{500} \ln (| 500 - y |) + C_{3} = \frac{1}{20} x + C_{4}$

Step 2.4

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

$\frac{1}{500} \ln (| y |) - \frac{1}{500} \ln (| 500 - y |) = \frac{1}{20} x + C$

Step 3

Solve for $y$ .

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Step 3.1

Simplify the expressions in the equation.

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Step 3.1.1

Simplify the left side.

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Step 3.1.1.1

Simplify each term.

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Step 3.1.1.1.1

Combine $\frac{1}{500}$ and $\ln (| y |)$ .

$\frac{\ln (| y |)}{500} - \frac{1}{500} \ln (| 500 - y |) = \frac{1}{20} x + C$

Step 3.1.1.1.2

Combine $\ln (| 500 - y |)$ and $\frac{1}{500}$ .

$\frac{\ln (| y |)}{500} - \frac{\ln (| 500 - y |)}{500} = \frac{1}{20} x + C$

Step 3.1.2

Simplify the right side.

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Step 3.1.2.1

Combine $\frac{1}{20}$ and $x$ .

$\frac{\ln (| y |)}{500} - \frac{\ln (| 500 - y |)}{500} = \frac{x}{20} + C$

Step 3.2

Multiply each term in $\frac{\ln (| y |)}{500} - \frac{\ln (| 500 - y |)}{500} = \frac{x}{20} + C$ by $500$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{\ln (| y |)}{500} - \frac{\ln (| 500 - y |)}{500} = \frac{x}{20} + C$ by $500$ .

$\frac{\ln (| y |)}{500} \cdot 500 - \frac{\ln (| 500 - y |)}{500} \cdot 500 = \frac{x}{20} \cdot 500 + C \cdot 500$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Cancel the common factor of $500$ .

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Step 3.2.2.1.1.1

Cancel the common factor.

$\frac{\ln (| y |)}{500} \cdot 500 - \frac{\ln (| 500 - y |)}{500} \cdot 500 = \frac{x}{20} \cdot 500 + C \cdot 500$

Step 3.2.2.1.1.2

Rewrite the expression.

$\ln (| y |) - \frac{\ln (| 500 - y |)}{500} \cdot 500 = \frac{x}{20} \cdot 500 + C \cdot 500$

Step 3.2.2.1.2

Cancel the common factor of $500$ .

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Step 3.2.2.1.2.1

Move the leading negative in $- \frac{\ln (| 500 - y |)}{500}$ into the numerator.

$\ln (| y |) + \frac{- \ln (| 500 - y |)}{500} \cdot 500 = \frac{x}{20} \cdot 500 + C \cdot 500$

Step 3.2.2.1.2.2

Cancel the common factor.

$\ln (| y |) + \frac{- \ln (| 500 - y |)}{500} \cdot 500 = \frac{x}{20} \cdot 500 + C \cdot 500$

Step 3.2.2.1.2.3

Rewrite the expression.

$\ln (| y |) - \ln (| 500 - y |) = \frac{x}{20} \cdot 500 + C \cdot 500$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Cancel the common factor of $20$ .

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Step 3.2.3.1.1.1

Factor $20$ out of $500$ .

$\ln (| y |) - \ln (| 500 - y |) = \frac{x}{20} \cdot (20 (25)) + C \cdot 500$

Step 3.2.3.1.1.2

Cancel the common factor.

$\ln (| y |) - \ln (| 500 - y |) = \frac{x}{20} \cdot (20 \cdot 25) + C \cdot 500$

Step 3.2.3.1.1.3

Rewrite the expression.

$\ln (| y |) - \ln (| 500 - y |) = x \cdot 25 + C \cdot 500$

Step 3.2.3.1.2

Move $25$ to the left of $x$ .

$\ln (| y |) - \ln (| 500 - y |) = 25 \cdot x + C \cdot 500$

Step 3.2.3.1.3

Move $500$ to the left of $C$ .

$\ln (| y |) - \ln (| 500 - y |) = 25 x + 500 C$

Step 3.3

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

$\ln (\frac{| y |}{| 500 - y |}) = 25 x + 500 C$

Step 3.4

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

$e^{\ln (\frac{| y |}{| 500 - y |})} = e^{25 x + 500 C}$

Step 3.5

Rewrite $\ln (\frac{| y |}{| 500 - y |}) = 25 x + 500 C$ 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 $b^{y} = x$ .

$e^{25 x + 500 C} = \frac{| y |}{| 500 - y |}$

Step 3.6

Solve for $y$ .

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Step 3.6.1

Rewrite the equation as $\frac{| y |}{| 500 - y |} = e^{25 x + 500 C}$ .

$\frac{| y |}{| 500 - y |} = e^{25 x + 500 C}$

Step 3.6.2

Multiply both sides by $| 500 - y |$ .

$\frac{| y |}{| 500 - y |} | 500 - y | = e^{25 x + 500 C} | 500 - y |$

Step 3.6.3

Simplify the left side.

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Step 3.6.3.1

Cancel the common factor of $| 500 - y |$ .

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Step 3.6.3.1.1

Cancel the common factor.

$\frac{| y |}{| 500 - y |} | 500 - y | = e^{25 x + 500 C} | 500 - y |$

Step 3.6.3.1.2

Rewrite the expression.

$| y | = e^{25 x + 500 C} | 500 - y |$

Step 3.6.4

Solve for $y$ .

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Step 3.6.4.1

Rewrite the equation as $e^{25 x + 500 C} | 500 - y | = | y |$ .

$e^{25 x + 500 C} | 500 - y | = | y |$

Step 3.6.4.2

Divide each term in $e^{25 x + 500 C} | 500 - y | = | y |$ by $e^{25 x + 500 C}$ and simplify.

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Step 3.6.4.2.1

Divide each term in $e^{25 x + 500 C} | 500 - y | = | y |$ by $e^{25 x + 500 C}$ .

$\frac{e^{25 x + 500 C} | 500 - y |}{e^{25 x + 500 C}} = \frac{| y |}{e^{25 x + 500 C}}$

Step 3.6.4.2.2

Simplify the left side.

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Step 3.6.4.2.2.1

Cancel the common factor of $e^{25 x + 500 C}$ .

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Step 3.6.4.2.2.1.1

Cancel the common factor.

$\frac{e^{25 x + 500 C} | 500 - y |}{e^{25 x + 500 C}} = \frac{| y |}{e^{25 x + 500 C}}$

Step 3.6.4.2.2.1.2

Divide $| 500 - y |$ by $1$ .

$| 500 - y | = \frac{| y |}{e^{25 x + 500 C}}$

Step 3.6.4.3

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

$500 - y = \frac{y}{e^{25 x + 500 C}}$

$500 - y = - \frac{y}{e^{25 x + 500 C}}$

$- (500 - y) = \frac{y}{e^{25 x + 500 C}}$

$- (500 - y) = - \frac{y}{e^{25 x + 500 C}}$

Step 3.6.4.4

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

$500 - y = \frac{y}{e^{25 x + 500 C}}$

$500 - y = - \frac{y}{e^{25 x + 500 C}}$

Step 3.6.4.5

Solve $500 - y = \frac{y}{e^{25 x + 500 C}}$ for $y$ .

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Step 3.6.4.5.1

Multiply both sides by $e^{25 x + 500 C}$ .

$(500 - y) e^{25 x + 500 C} = \frac{y}{e^{25 x + 500 C}} e^{25 x + 500 C}$

Step 3.6.4.5.2

Simplify.

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Step 3.6.4.5.2.1

Simplify the left side.

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Step 3.6.4.5.2.1.1

Simplify $(500 - y) e^{25 x + 500 C}$ .

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Step 3.6.4.5.2.1.1.1

Apply the distributive property.

$500 e^{25 x + 500 C} - y e^{25 x + 500 C} = \frac{y}{e^{25 x + 500 C}} e^{25 x + 500 C}$

Step 3.6.4.5.2.1.1.2

Simplify with commuting.

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Step 3.6.4.5.2.1.1.2.1

Reorder $25 x$ and $500 C$ .

$500 e^{500 C + 25 x} - y e^{25 x + 500 C} = \frac{y}{e^{25 x + 500 C}} e^{25 x + 500 C}$

Step 3.6.4.5.2.1.1.2.2

Reorder $25 x$ and $500 C$ .

$500 e^{500 C + 25 x} - y e^{500 C + 25 x} = \frac{y}{e^{25 x + 500 C}} e^{25 x + 500 C}$

Step 3.6.4.5.2.1.1.2.3

Reorder $500 e^{500 C + 25 x}$ and $- y e^{500 C + 25 x}$ .

$- y e^{500 C + 25 x} + 500 e^{500 C + 25 x} = \frac{y}{e^{25 x + 500 C}} e^{25 x + 500 C}$

Step 3.6.4.5.2.2

Simplify the right side.

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Step 3.6.4.5.2.2.1

Cancel the common factor of $e^{25 x + 500 C}$ .

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Step 3.6.4.5.2.2.1.1

Cancel the common factor.

$- y e^{500 C + 25 x} + 500 e^{500 C + 25 x} = \frac{y}{e^{25 x + 500 C}} e^{25 x + 500 C}$

Step 3.6.4.5.2.2.1.2

Rewrite the expression.

$- y e^{500 C + 25 x} + 500 e^{500 C + 25 x} = y$

Step 3.6.4.5.3

Solve for $y$ .

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Step 3.6.4.5.3.1

Subtract $y$ from both sides of the equation.

$- y e^{500 C + 25 x} + 500 e^{500 C + 25 x} - y = 0$

Step 3.6.4.5.3.2

Subtract $500 e^{500 C + 25 x}$ from both sides of the equation.

$- y e^{500 C + 25 x} - y = - 500 e^{500 C + 25 x}$

Step 3.6.4.5.3.3

Factor $y$ out of $- y e^{500 C + 25 x} - y$ .

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Step 3.6.4.5.3.3.1

Factor $y$ out of $- y e^{500 C + 25 x}$ .

$y (- 1 e^{500 C + 25 x}) - y = - 500 e^{500 C + 25 x}$

Step 3.6.4.5.3.3.2

Factor $y$ out of $- y$ .

$y (- 1 e^{500 C + 25 x}) + y \cdot - 1 = - 500 e^{500 C + 25 x}$

Step 3.6.4.5.3.3.3

Factor $y$ out of $y (- 1 e^{500 C + 25 x}) + y \cdot - 1$ .

$y (- 1 e^{500 C + 25 x} - 1) = - 500 e^{500 C + 25 x}$

Step 3.6.4.5.3.4

Rewrite $- 1 e^{500 C + 25 x}$ as $- e^{500 C + 25 x}$ .

$y (- e^{500 C + 25 x} - 1) = - 500 e^{500 C + 25 x}$

Step 3.6.4.5.3.5

Divide each term in $y (- e^{500 C + 25 x} - 1) = - 500 e^{500 C + 25 x}$ by $- e^{500 C + 25 x} - 1$ and simplify.

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Step 3.6.4.5.3.5.1

Divide each term in $y (- e^{500 C + 25 x} - 1) = - 500 e^{500 C + 25 x}$ by $- e^{500 C + 25 x} - 1$ .

$\frac{y (- e^{500 C + 25 x} - 1)}{- e^{500 C + 25 x} - 1} = \frac{- 500 e^{500 C + 25 x}}{- e^{500 C + 25 x} - 1}$

Step 3.6.4.5.3.5.2

Simplify the left side.

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Step 3.6.4.5.3.5.2.1

Cancel the common factor of $- e^{500 C + 25 x} - 1$ .

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Step 3.6.4.5.3.5.2.1.1

Cancel the common factor.

$\frac{y (- e^{500 C + 25 x} - 1)}{- e^{500 C + 25 x} - 1} = \frac{- 500 e^{500 C + 25 x}}{- e^{500 C + 25 x} - 1}$

Step 3.6.4.5.3.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 500 e^{500 C + 25 x}}{- e^{500 C + 25 x} - 1}$

Step 3.6.4.5.3.5.3

Simplify the right side.

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Step 3.6.4.5.3.5.3.1

Move the negative in front of the fraction.

$y = - \frac{500 e^{500 C + 25 x}}{- e^{500 C + 25 x} - 1}$

Step 3.6.4.5.3.5.3.2

Factor $- 1$ out of $- e^{500 C + 25 x}$ .

$y = - \frac{500 e^{500 C + 25 x}}{- e^{500 C + 25 x} - 1}$

Step 3.6.4.5.3.5.3.3

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

$y = - \frac{500 e^{500 C + 25 x}}{- e^{500 C + 25 x} - 1 (1)}$

Step 3.6.4.5.3.5.3.4

Factor $- 1$ out of $- e^{500 C + 25 x} - 1 (1)$ .

$y = - \frac{500 e^{500 C + 25 x}}{- (e^{500 C + 25 x} + 1)}$

Step 3.6.4.5.3.5.3.5

Simplify the expression.

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Step 3.6.4.5.3.5.3.5.1

Rewrite $- (e^{500 C + 25 x} + 1)$ as $- 1 (e^{500 C + 25 x} + 1)$ .

$y = - \frac{500 e^{500 C + 25 x}}{- 1 (e^{500 C + 25 x} + 1)}$

Step 3.6.4.5.3.5.3.5.2

Move the negative in front of the fraction.

$y = - - \frac{500 e^{500 C + 25 x}}{e^{500 C + 25 x} + 1}$

Step 3.6.4.5.3.5.3.5.3

Multiply $- 1$ by $- 1$ .

$y = 1 \frac{500 e^{500 C + 25 x}}{e^{500 C + 25 x} + 1}$

Step 3.6.4.5.3.5.3.5.4

Multiply $\frac{500 e^{500 C + 25 x}}{e^{500 C + 25 x} + 1}$ by $1$ .

$y = \frac{500 e^{500 C + 25 x}}{e^{500 C + 25 x} + 1}$

Step 3.6.4.6

Solve $500 - y = - \frac{y}{e^{25 x + 500 C}}$ for $y$ .

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Step 3.6.4.6.1

Multiply both sides by $e^{25 x + 500 C}$ .

$(500 - y) e^{25 x + 500 C} = - \frac{y}{e^{25 x + 500 C}} e^{25 x + 500 C}$

Step 3.6.4.6.2

Simplify.

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Step 3.6.4.6.2.1

Simplify the left side.

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Step 3.6.4.6.2.1.1

Simplify $(500 - y) e^{25 x + 500 C}$ .

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Step 3.6.4.6.2.1.1.1

Apply the distributive property.

$500 e^{25 x + 500 C} - y e^{25 x + 500 C} = - \frac{y}{e^{25 x + 500 C}} e^{25 x + 500 C}$

Step 3.6.4.6.2.1.1.2

Simplify with commuting.

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Step 3.6.4.6.2.1.1.2.1

Reorder $25 x$ and $500 C$ .

$500 e^{500 C + 25 x} - y e^{25 x + 500 C} = - \frac{y}{e^{25 x + 500 C}} e^{25 x + 500 C}$

Step 3.6.4.6.2.1.1.2.2

Reorder $25 x$ and $500 C$ .

$500 e^{500 C + 25 x} - y e^{500 C + 25 x} = - \frac{y}{e^{25 x + 500 C}} e^{25 x + 500 C}$

Step 3.6.4.6.2.1.1.2.3

Reorder $500 e^{500 C + 25 x}$ and $- y e^{500 C + 25 x}$ .

$- y e^{500 C + 25 x} + 500 e^{500 C + 25 x} = - \frac{y}{e^{25 x + 500 C}} e^{25 x + 500 C}$

Step 3.6.4.6.2.2

Simplify the right side.

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Step 3.6.4.6.2.2.1

Cancel the common factor of $e^{25 x + 500 C}$ .

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Step 3.6.4.6.2.2.1.1

Move the leading negative in $- \frac{y}{e^{25 x + 500 C}}$ into the numerator.

$- y e^{500 C + 25 x} + 500 e^{500 C + 25 x} = \frac{- y}{e^{25 x + 500 C}} e^{25 x + 500 C}$

Step 3.6.4.6.2.2.1.2

Cancel the common factor.

$- y e^{500 C + 25 x} + 500 e^{500 C + 25 x} = \frac{- y}{e^{25 x + 500 C}} e^{25 x + 500 C}$

Step 3.6.4.6.2.2.1.3

Rewrite the expression.

$- y e^{500 C + 25 x} + 500 e^{500 C + 25 x} = - y$

Step 3.6.4.6.3

Solve for $y$ .

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Step 3.6.4.6.3.1

Add $y$ to both sides of the equation.

$- y e^{500 C + 25 x} + 500 e^{500 C + 25 x} + y = 0$

Step 3.6.4.6.3.2

Subtract $500 e^{500 C + 25 x}$ from both sides of the equation.

$- y e^{500 C + 25 x} + y = - 500 e^{500 C + 25 x}$

Step 3.6.4.6.3.3

Factor $y$ out of $- y e^{500 C + 25 x} + y$ .

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Step 3.6.4.6.3.3.1

Factor $y$ out of $- y e^{500 C + 25 x}$ .

$y (- 1 e^{500 C + 25 x}) + y = - 500 e^{500 C + 25 x}$

Step 3.6.4.6.3.3.2

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

$y (- 1 e^{500 C + 25 x}) + y = - 500 e^{500 C + 25 x}$

Step 3.6.4.6.3.3.3

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

$y (- 1 e^{500 C + 25 x}) + y \cdot 1 = - 500 e^{500 C + 25 x}$

Step 3.6.4.6.3.3.4

Factor $y$ out of $y (- 1 e^{500 C + 25 x}) + y \cdot 1$ .

$y (- 1 e^{500 C + 25 x} + 1) = - 500 e^{500 C + 25 x}$

Step 3.6.4.6.3.4

Rewrite $- 1 e^{500 C + 25 x}$ as $- e^{500 C + 25 x}$ .

$y (- e^{500 C + 25 x} + 1) = - 500 e^{500 C + 25 x}$

Step 3.6.4.6.3.5

Divide each term in $y (- e^{500 C + 25 x} + 1) = - 500 e^{500 C + 25 x}$ by $- e^{500 C + 25 x} + 1$ and simplify.

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Step 3.6.4.6.3.5.1

Divide each term in $y (- e^{500 C + 25 x} + 1) = - 500 e^{500 C + 25 x}$ by $- e^{500 C + 25 x} + 1$ .

$\frac{y (- e^{500 C + 25 x} + 1)}{- e^{500 C + 25 x} + 1} = \frac{- 500 e^{500 C + 25 x}}{- e^{500 C + 25 x} + 1}$

Step 3.6.4.6.3.5.2

Simplify the left side.

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Step 3.6.4.6.3.5.2.1

Cancel the common factor of $- e^{500 C + 25 x} + 1$ .

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Step 3.6.4.6.3.5.2.1.1

Cancel the common factor.

$\frac{y (- e^{500 C + 25 x} + 1)}{- e^{500 C + 25 x} + 1} = \frac{- 500 e^{500 C + 25 x}}{- e^{500 C + 25 x} + 1}$

Step 3.6.4.6.3.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 500 e^{500 C + 25 x}}{- e^{500 C + 25 x} + 1}$

Step 3.6.4.6.3.5.3

Simplify the right side.

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Step 3.6.4.6.3.5.3.1

Move the negative in front of the fraction.

$y = - \frac{500 e^{500 C + 25 x}}{- e^{500 C + 25 x} + 1}$

Step 3.6.4.6.3.5.3.2

Factor $- 1$ out of $- e^{500 C + 25 x}$ .

$y = - \frac{500 e^{500 C + 25 x}}{- e^{500 C + 25 x} + 1}$

Step 3.6.4.6.3.5.3.3

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

$y = - \frac{500 e^{500 C + 25 x}}{- e^{500 C + 25 x} - 1 (- 1)}$

Step 3.6.4.6.3.5.3.4

Factor $- 1$ out of $- e^{500 C + 25 x} - 1 (- 1)$ .

$y = - \frac{500 e^{500 C + 25 x}}{- (e^{500 C + 25 x} - 1)}$

Step 3.6.4.6.3.5.3.5

Simplify the expression.

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Step 3.6.4.6.3.5.3.5.1

Rewrite $- (e^{500 C + 25 x} - 1)$ as $- 1 (e^{500 C + 25 x} - 1)$ .

$y = - \frac{500 e^{500 C + 25 x}}{- 1 (e^{500 C + 25 x} - 1)}$

Step 3.6.4.6.3.5.3.5.2

Move the negative in front of the fraction.

$y = - - \frac{500 e^{500 C + 25 x}}{e^{500 C + 25 x} - 1}$

Step 3.6.4.6.3.5.3.5.3

Multiply $- 1$ by $- 1$ .

$y = 1 \frac{500 e^{500 C + 25 x}}{e^{500 C + 25 x} - 1}$

Step 3.6.4.6.3.5.3.5.4

Multiply $\frac{500 e^{500 C + 25 x}}{e^{500 C + 25 x} - 1}$ by $1$ .

$y = \frac{500 e^{500 C + 25 x}}{e^{500 C + 25 x} - 1}$

Step 3.6.4.7

List all of the solutions.

$y = \frac{500 e^{500 C + 25 x}}{e^{500 C + 25 x} + 1}$

$y = \frac{500 e^{500 C + 25 x}}{e^{500 C + 25 x} - 1}$

$y = \frac{500 e^{500 C + 25 x}}{e^{500 C + 25 x} + 1}$

$y = \frac{500 e^{500 C + 25 x}}{e^{500 C + 25 x} - 1}$

$y = \frac{500 e^{500 C + 25 x}}{e^{500 C + 25 x} + 1}$

$y = \frac{500 e^{500 C + 25 x}}{e^{500 C + 25 x} - 1}$

$y = \frac{500 e^{500 C + 25 x}}{e^{500 C + 25 x} + 1}$

$y = \frac{500 e^{500 C + 25 x}}{e^{500 C + 25 x} - 1}$

Step 4

Group the constant terms together.

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Step 4.1

Simplify the constant of integration.

$y = \frac{500 e^{C + 25 x}}{e^{C + 25 x} + 1}$

$y = \frac{500 e^{C + 25 x}}{e^{C + 25 x} - 1}$

Step 4.2

Reorder terms.

$y = \frac{500 e^{25 x + C}}{e^{C + 25 x} + 1}$

$y = \frac{500 e^{C + 25 x}}{e^{C + 25 x} - 1}$

Step 4.3

Rewrite $e^{25 x + C}$ as $e^{25 x} e^{C}$ .

$y = \frac{500 (e^{25 x} e^{C})}{e^{C + 25 x} + 1}$

$y = \frac{500 e^{C + 25 x}}{e^{C + 25 x} - 1}$

Step 4.4

Reorder $e^{25 x}$ and $e^{C}$ .

$y = \frac{500 (e^{C} e^{25 x})}{e^{C + 25 x} + 1}$

$y = \frac{500 e^{C + 25 x}}{e^{C + 25 x} - 1}$

Step 4.5

Reorder terms.

$y = \frac{500 (e^{C} e^{25 x})}{e^{25 x + C} + 1}$

$y = \frac{500 e^{C + 25 x}}{e^{C + 25 x} - 1}$

Step 4.6

Rewrite $e^{25 x + C}$ as $e^{25 x} e^{C}$ .

$y = \frac{500 (e^{C} e^{25 x})}{e^{25 x} e^{C} + 1}$

$y = \frac{500 e^{C + 25 x}}{e^{C + 25 x} - 1}$

Step 4.7

Reorder $e^{25 x}$ and $e^{C}$ .