Calculus Examples

$\frac{d y}{d x} + \frac{2}{20 - x} y = 4$

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{2}{20 - x}$ .

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Step 1.1

Set up the integration.

$e^{\int \frac{2}{20 - x} d x}$

Step 1.2

Integrate $\frac{2}{20 - x}$ .

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Step 1.2.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$e^{2 \int \frac{1}{20 - x} d x}$

Step 1.2.2

Let $u = 20 - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.2.2.1

Let $u = 20 - x$ . Find $\frac{d u}{d x}$ .

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Step 1.2.2.1.1

Rewrite.

$\frac{- 1}{1}$

Step 1.2.2.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 1.2.2.2

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

$e^{2 \int \frac{- 1}{u} d u}$

Step 1.2.3

Move the negative in front of the fraction.

$e^{2 \int - \frac{1}{u} d u}$

Step 1.2.4

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

$e^{2 (- \int \frac{1}{u} d u)}$

Step 1.2.5

Multiply $- 1$ by $2$ .

$e^{- 2 \int \frac{1}{u} d u}$

Step 1.2.6

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

$e^{- 2 (\ln (| u |) + C)}$

Step 1.2.7

Simplify.

$e^{- 2 \ln (| u |) + C}$

Step 1.2.8

Replace all occurrences of $u$ with $20 - x$ .

$e^{- 2 \ln (| 20 - x |) + C}$

Step 1.3

Remove the constant of integration.

$e^{- 2 \ln (20 - x)}$

Step 1.4

Use the logarithmic power rule.

$e^{\ln ({(20 - x)}^{- 2})}$

Step 1.5

Exponentiation and log are inverse functions.

${(20 - x)}^{- 2}$

Step 1.6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(20 - x)}^{2}}$

Step 2

Multiply each term by the integrating factor $\frac{1}{{(20 - x)}^{2}}$ .

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Step 2.1

Multiply each term by $\frac{1}{{(20 - x)}^{2}}$ .

$\frac{1}{{(20 - x)}^{2}} \frac{d y}{d x} + \frac{1}{{(20 - x)}^{2}} (\frac{2}{20 - x} y) = \frac{1}{{(20 - x)}^{2}} \cdot 4$

Step 2.2

Simplify each term.

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Step 2.2.1

Combine $\frac{1}{{(20 - x)}^{2}}$ and $\frac{d y}{d x}$ .

$\frac{\frac{d y}{d x}}{{(20 - x)}^{2}} + \frac{1}{{(20 - x)}^{2}} (\frac{2}{20 - x} y) = \frac{1}{{(20 - x)}^{2}} \cdot 4$

Step 2.2.2

Combine $\frac{2}{20 - x}$ and $y$ .

$\frac{\frac{d y}{d x}}{{(20 - x)}^{2}} + \frac{1}{{(20 - x)}^{2}} \cdot \frac{2 y}{20 - x} = \frac{1}{{(20 - x)}^{2}} \cdot 4$

Step 2.2.3

Multiply $\frac{1}{{(20 - x)}^{2}} \cdot \frac{2 y}{20 - x}$ .

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Step 2.2.3.1

Multiply $\frac{1}{{(20 - x)}^{2}}$ by $\frac{2 y}{20 - x}$ .

$\frac{\frac{d y}{d x}}{{(20 - x)}^{2}} + \frac{2 y}{{(20 - x)}^{2} (20 - x)} = \frac{1}{{(20 - x)}^{2}} \cdot 4$

Step 2.2.3.2

Multiply ${(20 - x)}^{2}$ by $20 - x$ by adding the exponents.

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Step 2.2.3.2.1

Multiply ${(20 - x)}^{2}$ by $20 - x$ .

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Step 2.2.3.2.1.1

Raise $20 - x$ to the power of $1$ .

$\frac{\frac{d y}{d x}}{{(20 - x)}^{2}} + \frac{2 y}{{(20 - x)}^{2} {(20 - x)}^{1}} = \frac{1}{{(20 - x)}^{2}} \cdot 4$

Step 2.2.3.2.1.2

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

$\frac{\frac{d y}{d x}}{{(20 - x)}^{2}} + \frac{2 y}{{(20 - x)}^{2 + 1}} = \frac{1}{{(20 - x)}^{2}} \cdot 4$

Step 2.2.3.2.2

Add $2$ and $1$ .

$\frac{\frac{d y}{d x}}{{(20 - x)}^{2}} + \frac{2 y}{{(20 - x)}^{3}} = \frac{1}{{(20 - x)}^{2}} \cdot 4$

Step 2.3

Combine $\frac{1}{{(20 - x)}^{2}}$ and $4$ .

$\frac{\frac{d y}{d x}}{{(20 - x)}^{2}} + \frac{2 y}{{(20 - x)}^{3}} = \frac{4}{{(20 - x)}^{2}}$

Step 3

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [\frac{1}{{(20 - x)}^{2}} y] = \frac{4}{{(20 - x)}^{2}}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{{(20 - x)}^{2}} y] d x = \int \frac{4}{{(20 - x)}^{2}} d x$

Step 5

Integrate the left side.

$\frac{1}{{(20 - x)}^{2}} y = \int \frac{4}{{(20 - x)}^{2}} d x$

Step 6

Integrate the right side.

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Step 6.1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$\frac{1}{{(20 - x)}^{2}} y = 4 \int \frac{1}{{(20 - x)}^{2}} d x$

Step 6.2

Let $u = 20 - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.2.1

Let $u = 20 - x$ . Find $\frac{d u}{d x}$ .

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Step 6.2.1.1

Rewrite.

$\frac{- 1}{1}$

Step 6.2.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 6.2.2

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

$\frac{1}{{(20 - x)}^{2}} y = 4 \int \frac{- 1}{u^{2}} d u$

Step 6.3

Move the negative in front of the fraction.

$\frac{1}{{(20 - x)}^{2}} y = 4 \int - \frac{1}{u^{2}} d u$

Step 6.4

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

$\frac{1}{{(20 - x)}^{2}} y = 4 (- \int \frac{1}{u^{2}} d u)$

Step 6.5

Simplify the expression.

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Step 6.5.1

Multiply $- 1$ by $4$ .

$\frac{1}{{(20 - x)}^{2}} y = - 4 \int \frac{1}{u^{2}} d u$

Step 6.5.2

Move $u^{2}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{{(20 - x)}^{2}} y = - 4 \int {(u^{2})}^{- 1} d u$

Step 6.5.3

Multiply the exponents in ${(u^{2})}^{- 1}$ .

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Step 6.5.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{{(20 - x)}^{2}} y = - 4 \int u^{2 \cdot - 1} d u$

Step 6.5.3.2

Multiply $2$ by $- 1$ .

$\frac{1}{{(20 - x)}^{2}} y = - 4 \int u^{- 2} d u$

Step 6.6

By the Power Rule, the integral of $u^{- 2}$ with respect to $u$ is $- u^{- 1}$ .

$\frac{1}{{(20 - x)}^{2}} y = - 4 (- u^{- 1} + C)$

Step 6.7

Simplify.

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Step 6.7.1

Rewrite $- 4 (- u^{- 1} + C)$ as $- 4 (- \frac{1}{u}) + C$ .

$\frac{1}{{(20 - x)}^{2}} y = - 4 (- \frac{1}{u}) + C$

Step 6.7.2

Simplify.

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Step 6.7.2.1

Multiply $- 1$ by $- 4$ .

$\frac{1}{{(20 - x)}^{2}} y = 4 \frac{1}{u} + C$

Step 6.7.2.2

Combine $4$ and $\frac{1}{u}$ .

$\frac{1}{{(20 - x)}^{2}} y = \frac{4}{u} + C$

Step 6.8

Replace all occurrences of $u$ with $20 - x$ .

$\frac{1}{{(20 - x)}^{2}} y = \frac{4}{20 - x} + C$

Step 7

Solve for $y$ .

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Step 7.1

Move all terms containing variables to the left side of the equation.

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Step 7.1.1

Subtract $\frac{4}{20 - x}$ from both sides of the equation.

$\frac{1}{{(20 - x)}^{2}} y - \frac{4}{20 - x} = C$

Step 7.1.2

Subtract $C$ from both sides of the equation.

$\frac{1}{{(20 - x)}^{2}} y - \frac{4}{20 - x} - C = 0$

Step 7.1.3

Combine $\frac{1}{{(20 - x)}^{2}}$ and $y$ .

$\frac{y}{{(20 - x)}^{2}} - \frac{4}{20 - x} - C = 0$

Step 7.1.4

To write $- \frac{4}{20 - x}$ as a fraction with a common denominator, multiply by $\frac{20 - x}{20 - x}$ .

$\frac{y}{{(20 - x)}^{2}} - \frac{4}{20 - x} \cdot \frac{20 - x}{20 - x} - C = 0$

Step 7.1.5

Write each expression with a common denominator of ${(20 - x)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1.5.1

Multiply $\frac{4}{20 - x}$ by $\frac{20 - x}{20 - x}$ .

$\frac{y}{{(20 - x)}^{2}} - \frac{4 (20 - x)}{(20 - x) (20 - x)} - C = 0$

Step 7.1.5.2

Raise $20 - x$ to the power of $1$ .

$\frac{y}{{(20 - x)}^{2}} - \frac{4 (20 - x)}{{(20 - x)}^{1} (20 - x)} - C = 0$

Step 7.1.5.3

Raise $20 - x$ to the power of $1$ .

$\frac{y}{{(20 - x)}^{2}} - \frac{4 (20 - x)}{{(20 - x)}^{1} {(20 - x)}^{1}} - C = 0$

Step 7.1.5.4

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

$\frac{y}{{(20 - x)}^{2}} - \frac{4 (20 - x)}{{(20 - x)}^{1 + 1}} - C = 0$

Step 7.1.5.5

Add $1$ and $1$ .

$\frac{y}{{(20 - x)}^{2}} - \frac{4 (20 - x)}{{(20 - x)}^{2}} - C = 0$

Step 7.1.6

Combine the numerators over the common denominator.

$\frac{y - 4 (20 - x)}{{(20 - x)}^{2}} - C = 0$

Step 7.1.7

Simplify the numerator.

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Step 7.1.7.1

Apply the distributive property.

$\frac{y - 4 \cdot 20 - 4 (- x)}{{(20 - x)}^{2}} - C = 0$

Step 7.1.7.2

Multiply $- 4$ by $20$ .

$\frac{y - 80 - 4 (- x)}{{(20 - x)}^{2}} - C = 0$

Step 7.1.7.3

Multiply $- 1$ by $- 4$ .

$\frac{y - 80 + 4 x}{{(20 - x)}^{2}} - C = 0$

Step 7.1.8

To write $- C$ as a fraction with a common denominator, multiply by $\frac{{(20 - x)}^{2}}{{(20 - x)}^{2}}$ .

$\frac{y - 80 + 4 x}{{(20 - x)}^{2}} - C \cdot \frac{{(20 - x)}^{2}}{{(20 - x)}^{2}} = 0$

Step 7.1.9

Combine $- C$ and $\frac{{(20 - x)}^{2}}{{(20 - x)}^{2}}$ .

$\frac{y - 80 + 4 x}{{(20 - x)}^{2}} + \frac{- C {(20 - x)}^{2}}{{(20 - x)}^{2}} = 0$

Step 7.1.10

Combine the numerators over the common denominator.

$\frac{y - 80 + 4 x - C {(20 - x)}^{2}}{{(20 - x)}^{2}} = 0$

Step 7.1.11

Simplify the numerator.

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Step 7.1.11.1

Rewrite ${(20 - x)}^{2}$ as $(20 - x) (20 - x)$ .

$\frac{y - 80 + 4 x - C ((20 - x) (20 - x))}{{(20 - x)}^{2}} = 0$

Step 7.1.11.2

Expand $(20 - x) (20 - x)$ using the FOIL Method.

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Step 7.1.11.2.1

Apply the distributive property.

$\frac{y - 80 + 4 x - C (20 (20 - x) - x (20 - x))}{{(20 - x)}^{2}} = 0$

Step 7.1.11.2.2

Apply the distributive property.

$\frac{y - 80 + 4 x - C (20 \cdot 20 + 20 (- x) - x (20 - x))}{{(20 - x)}^{2}} = 0$

Step 7.1.11.2.3

Apply the distributive property.

$\frac{y - 80 + 4 x - C (20 \cdot 20 + 20 (- x) - x \cdot 20 - x (- x))}{{(20 - x)}^{2}} = 0$

Step 7.1.11.3

Simplify and combine like terms.

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Step 7.1.11.3.1

Simplify each term.

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Step 7.1.11.3.1.1

Multiply $20$ by $20$ .

$\frac{y - 80 + 4 x - C (400 + 20 (- x) - x \cdot 20 - x (- x))}{{(20 - x)}^{2}} = 0$

Step 7.1.11.3.1.2

Multiply $- 1$ by $20$ .

$\frac{y - 80 + 4 x - C (400 - 20 x - x \cdot 20 - x (- x))}{{(20 - x)}^{2}} = 0$

Step 7.1.11.3.1.3

Multiply $20$ by $- 1$ .

$\frac{y - 80 + 4 x - C (400 - 20 x - 20 x - x (- x))}{{(20 - x)}^{2}} = 0$

Step 7.1.11.3.1.4

Rewrite using the commutative property of multiplication.

$\frac{y - 80 + 4 x - C (400 - 20 x - 20 x - 1 \cdot - 1 x \cdot x)}{{(20 - x)}^{2}} = 0$

Step 7.1.11.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 7.1.11.3.1.5.1

Move $x$ .

$\frac{y - 80 + 4 x - C (400 - 20 x - 20 x - 1 \cdot - 1 (x \cdot x))}{{(20 - x)}^{2}} = 0$

Step 7.1.11.3.1.5.2

Multiply $x$ by $x$ .

$\frac{y - 80 + 4 x - C (400 - 20 x - 20 x - 1 \cdot - 1 x^{2})}{{(20 - x)}^{2}} = 0$

Step 7.1.11.3.1.6

Multiply $- 1$ by $- 1$ .

$\frac{y - 80 + 4 x - C (400 - 20 x - 20 x + 1 x^{2})}{{(20 - x)}^{2}} = 0$

Step 7.1.11.3.1.7

Multiply $x^{2}$ by $1$ .

$\frac{y - 80 + 4 x - C (400 - 20 x - 20 x + x^{2})}{{(20 - x)}^{2}} = 0$

Step 7.1.11.3.2

Subtract $20 x$ from $- 20 x$ .

$\frac{y - 80 + 4 x - C (400 - 40 x + x^{2})}{{(20 - x)}^{2}} = 0$

Step 7.1.11.4

Apply the distributive property.

$\frac{y - 80 + 4 x - C \cdot 400 - C (- 40 x) - C x^{2}}{{(20 - x)}^{2}} = 0$

Step 7.1.11.5

Simplify.

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Step 7.1.11.5.1

Multiply $400$ by $- 1$ .

$\frac{y - 80 + 4 x - 400 C - C (- 40 x) - C x^{2}}{{(20 - x)}^{2}} = 0$

Step 7.1.11.5.2

Rewrite using the commutative property of multiplication.

$\frac{y - 80 + 4 x - 400 C - 1 \cdot - 40 C x - C x^{2}}{{(20 - x)}^{2}} = 0$

Step 7.1.11.6

Multiply $- 1$ by $- 40$ .

$\frac{y - 80 + 4 x - 400 C + 40 C x - C x^{2}}{{(20 - x)}^{2}} = 0$

Step 7.2

Set the numerator equal to zero.

$y - 80 + 4 x - 400 C + 40 C x - C x^{2} = 0$

Step 7.3

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

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Step 7.3.1

Add $80$ to both sides of the equation.

$y + 4 x - 400 C + 40 C x - C x^{2} = 80$

Step 7.3.2

Subtract $4 x$ from both sides of the equation.

$y - 400 C + 40 C x - C x^{2} = 80 - 4 x$

Step 7.3.3

Add $400 C$ to both sides of the equation.

$y + 40 C x - C x^{2} = 80 - 4 x + 400 C$

Step 7.3.4

Subtract $40 C x$ from both sides of the equation.

$y - C x^{2} = 80 - 4 x + 400 C - 40 C x$

Step 7.3.5

Add $C x^{2}$ to both sides of the equation.

$y = 80 - 4 x + 400 C - 40 C x + C x^{2}$