Calculus Examples

$\frac{d y}{d x} = 2 - 0.02 y$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{2 - 0.02 y}$ .

$\frac{1}{2 - 0.02 y} \frac{d y}{d x} = \frac{1}{2 - 0.02 y} (2 - 0.02 y)$

Step 1.2

Cancel the common factor of $2 - 0.02 y$ .

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Step 1.2.1

Cancel the common factor.

$\frac{1}{2 - 0.02 y} \frac{d y}{d x} = \frac{1}{2 - 0.02 y} (2 - 0.02 y)$

Step 1.2.2

Rewrite the expression.

$\frac{1}{2 - 0.02 y} \frac{d y}{d x} = 1$

Step 1.3

Rewrite the equation.

$\frac{1}{2 - 0.02 y} d y = 1 d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{2 - 0.02 y} d y = \int d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u = 2 - 0.02 y$ . Then $d u = - 0.02 d y$ , so $\frac{1}{- 0.02} d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1.1

Let $u = 2 - 0.02 y$ . Find $\frac{d u}{d y}$ .

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Step 2.2.1.1.1

Rewrite.

$\frac{- 0.02}{1}$

Step 2.2.1.1.2

Divide $- 0.02$ by $1$ .

$- 0.02$

Step 2.2.1.2

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

$\int \frac{- 50}{u} d u = \int d x$

Step 2.2.2

Split the fraction into multiple fractions.

$\int - \frac{50}{u} d u = \int d x$

Step 2.2.3

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

$- \int \frac{50}{u} d u = \int d x$

Step 2.2.4

Since $50$ is constant with respect to $u$ , move $50$ out of the integral.

$- (50 \int \frac{1}{u} d u) = \int d x$

Step 2.2.5

Multiply $50$ by $- 1$ .

$- 50 \int \frac{1}{u} d u = \int d x$

Step 2.2.6

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

$- 50 (\ln (| u |) + C_{1}) = \int d x$

Step 2.2.7

Simplify.

$- 50 \ln (| u |) + C_{1} = \int d x$

Step 2.2.8

Replace all occurrences of $u$ with $2 - 0.02 y$ .

$- 50 \ln (| 2 - 0.02 y |) + C_{1} = \int d x$

Step 2.3

Apply the constant rule.

$- 50 \ln (| 2 - 0.02 y |) + C_{1} = x + C_{2}$

Step 2.4

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

$- 50 \ln (| 2 - 0.02 y |) = x + C$

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $- 50 \ln (| 2 - 0.02 y |) = x + C$ by $- 50$ and simplify.

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Step 3.1.1

Divide each term in $- 50 \ln (| 2 - 0.02 y |) = x + C$ by $- 50$ .

$\frac{- 50 \ln (| 2 - 0.02 y |)}{- 50} = \frac{x}{- 50} + \frac{C}{- 50}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $- 50$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{- 50 \ln (| 2 - 0.02 y |)}{- 50} = \frac{x}{- 50} + \frac{C}{- 50}$

Step 3.1.2.1.2

Divide $\ln (| 2 - 0.02 y |)$ by $1$ .

$\ln (| 2 - 0.02 y |) = \frac{x}{- 50} + \frac{C}{- 50}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Move the negative in front of the fraction.

$\ln (| 2 - 0.02 y |) = - \frac{x}{50} + \frac{C}{- 50}$

Step 3.1.3.1.2

Move the negative in front of the fraction.

$\ln (| 2 - 0.02 y |) = - \frac{x}{50} - \frac{C}{50}$

Step 3.2

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

$e^{\ln (| 2 - 0.02 y |)} = e^{- \frac{x}{50} - \frac{C}{50}}$

Step 3.3

Rewrite $\ln (| 2 - 0.02 y |) = - \frac{x}{50} - \frac{C}{50}$ 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 $b^{y} = x$ .

$e^{- \frac{x}{50} - \frac{C}{50}} = | 2 - 0.02 y |$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Rewrite the equation as $| 2 - 0.02 y | = e^{- \frac{x}{50} - \frac{C}{50}}$ .

$| 2 - 0.02 y | = e^{- \frac{x}{50} - \frac{C}{50}}$

Step 3.4.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$2 - 0.02 y = \pm e^{- \frac{x}{50} - \frac{C}{50}}$

Step 3.4.3

Subtract $2$ from both sides of the equation.

$- 0.02 y = \pm e^{- \frac{x}{50} - \frac{C}{50}} - 2$

Step 3.4.4

Divide each term in $- 0.02 y = \pm e^{- \frac{x}{50} - \frac{C}{50}} - 2$ by $- 0.02$ and simplify.

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Step 3.4.4.1

Divide each term in $- 0.02 y = \pm e^{- \frac{x}{50} - \frac{C}{50}} - 2$ by $- 0.02$ .

$\frac{- 0.02 y}{- 0.02} = \frac{\pm e^{- \frac{x}{50} - \frac{C}{50}}}{- 0.02} + \frac{- 2}{- 0.02}$

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $- 0.02$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{- 0.02 y}{- 0.02} = \frac{\pm e^{- \frac{x}{50} - \frac{C}{50}}}{- 0.02} + \frac{- 2}{- 0.02}$

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{\pm e^{- \frac{x}{50} - \frac{C}{50}}}{- 0.02} + \frac{- 2}{- 0.02}$

Step 3.4.4.3

Simplify the right side.

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Step 3.4.4.3.1

Simplify each term.

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Step 3.4.4.3.1.1

Combine the numerators over the common denominator.

$y = \frac{\pm e^{\frac{- x - C}{50}}}{- 0.02} + \frac{- 2}{- 0.02}$

Step 3.4.4.3.1.2

Simplify $\frac{\pm e^{\frac{- x - C}{50}}}{- 0.02}$ .

$y = \frac{\pm e^{\frac{- x - C}{50}}}{0.02} + \frac{- 2}{- 0.02}$

Step 3.4.4.3.1.3

Multiply by $1$ .

$y = \frac{1 (\pm e^{\frac{- x - C}{50}})}{0.02} + \frac{- 2}{- 0.02}$

Step 3.4.4.3.1.4

Factor $0.02$ out of $0.02$ .

$y = \frac{1 (\pm e^{\frac{- x - C}{50}})}{0.02 (1)} + \frac{- 2}{- 0.02}$

Step 3.4.4.3.1.5

Separate fractions.

$y = \frac{1}{0.02} \cdot \frac{\pm e^{\frac{- x - C}{50}}}{1} + \frac{- 2}{- 0.02}$

Step 3.4.4.3.1.6

Divide $1$ by $0.02$ .

$y = 50 \frac{\pm e^{\frac{- x - C}{50}}}{1} + \frac{- 2}{- 0.02}$

Step 3.4.4.3.1.7

Divide $\pm e^{\frac{- x - C}{50}}$ by $1$ .

$y = 50 (\pm e^{\frac{- x - C}{50}}) + \frac{- 2}{- 0.02}$

Step 3.4.4.3.1.8

Divide $- 2$ by $- 0.02$ .

$y = 50 (\pm e^{\frac{- x - C}{50}}) + 100$

Step 4

Simplify the constant of integration.

$y = 50 (\pm e^{\frac{- x + C}{50}}) + 100$