Calculus Examples

$\frac{d y}{d x} + \frac{y}{x} = y^{9}$

Step 1

Rewrite the differential equation to fit the Bernoulli technique.

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Step 1.1

Factor $y$ out of $\frac{y}{x}$ .

$\frac{d y}{d x} + y \frac{1}{x} = y^{9}$

Step 1.2

Reorder $y$ and $\frac{1}{x}$ .

$\frac{d y}{d x} + \frac{1}{x} y = y^{9}$

Step 2

To solve the differential equation, let $v = y^{1 - n}$ where $n$ is the exponent of $y^{9}$ .

$v = y^{- 8}$

Step 3

Solve the equation for $y$ .

$y = v^{- \frac{1}{8}}$

Step 4

Take the derivative of $y$ with respect to $x$ .

$y' = v^{- \frac{1}{8}}$

Step 5

Take the derivative of $v^{- \frac{1}{8}}$ with respect to $x$ .

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Step 5.1

Take the derivative of $v^{- \frac{1}{8}}$ .

$y' = \frac{d}{d x} [v^{- \frac{1}{8}}]$

Step 5.2

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

$y' = \frac{d}{d x} [\frac{1}{v^{\frac{1}{8}}}]$

Step 5.3

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 1$ and $g (x) = v^{\frac{1}{8}}$ .

$y' = \frac{v^{\frac{1}{8}} \frac{d}{d x} [1] - 1 \cdot 1 \frac{d}{d x} [v^{\frac{1}{8}}]}{{(v^{\frac{1}{8}})}^{2}}$

Step 5.4

Differentiate using the Constant Rule.

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Step 5.4.1

Multiply $- 1$ by $1$ .

$y' = \frac{v^{\frac{1}{8}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{8}}]}{{(v^{\frac{1}{8}})}^{2}}$

Step 5.4.2

Multiply the exponents in ${(v^{\frac{1}{8}})}^{2}$ .

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Step 5.4.2.1

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

$y' = \frac{v^{\frac{1}{8}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{8}}]}{v^{\frac{1}{8} \cdot 2}}$

Step 5.4.2.2

Cancel the common factor of $2$ .

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Step 5.4.2.2.1

Factor $2$ out of $8$ .

$y' = \frac{v^{\frac{1}{8}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{8}}]}{v^{\frac{1}{2 (4)} \cdot 2}}$

Step 5.4.2.2.2

Cancel the common factor.

$y' = \frac{v^{\frac{1}{8}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{8}}]}{v^{\frac{1}{2 \cdot 4} \cdot 2}}$

Step 5.4.2.2.3

Rewrite the expression.

$y' = \frac{v^{\frac{1}{8}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{8}}]}{v^{\frac{1}{4}}}$

Step 5.4.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$y' = \frac{v^{\frac{1}{8}} \cdot 0 - \frac{d}{d x} [v^{\frac{1}{8}}]}{v^{\frac{1}{4}}}$

Step 5.4.4

Simplify the expression.

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Step 5.4.4.1

Multiply $v^{\frac{1}{8}}$ by $0$ .

$y' = \frac{0 - \frac{d}{d x} [v^{\frac{1}{8}}]}{v^{\frac{1}{4}}}$

Step 5.4.4.2

Subtract $\frac{d}{d x} [v^{\frac{1}{8}}]$ from $0$ .

$y' = \frac{- \frac{d}{d x} [v^{\frac{1}{8}}]}{v^{\frac{1}{4}}}$

Step 5.4.4.3

Move the negative in front of the fraction.

$y' = - \frac{\frac{d}{d x} [v^{\frac{1}{8}}]}{v^{\frac{1}{4}}}$

Step 5.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{8}}$ and $g (x) = v$ .

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Step 5.5.1

To apply the Chain Rule, set $u$ as $v$ .

$y' = - \frac{\frac{d}{d u} [u^{\frac{1}{8}}] \frac{d}{d x} [v]}{v^{\frac{1}{4}}}$

Step 5.5.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{8}$ .

$y' = - \frac{\frac{1}{8} u^{\frac{1}{8} - 1} \frac{d}{d x} [v]}{v^{\frac{1}{4}}}$

Step 5.5.3

Replace all occurrences of $u$ with $v$ .

$y' = - \frac{\frac{1}{8} v^{\frac{1}{8} - 1} \frac{d}{d x} [v]}{v^{\frac{1}{4}}}$

Step 5.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y' = - \frac{\frac{1}{8} v^{\frac{1}{8} - 1 \cdot \frac{8}{8}} \frac{d}{d x} [v]}{v^{\frac{1}{4}}}$

Step 5.7

Combine $- 1$ and $\frac{8}{8}$ .

$y' = - \frac{\frac{1}{8} v^{\frac{1}{8} + \frac{- 1 \cdot 8}{8}} \frac{d}{d x} [v]}{v^{\frac{1}{4}}}$

Step 5.8

Combine the numerators over the common denominator.

$y' = - \frac{\frac{1}{8} v^{\frac{1 - 1 \cdot 8}{8}} \frac{d}{d x} [v]}{v^{\frac{1}{4}}}$

Step 5.9

Simplify the numerator.

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Step 5.9.1

Multiply $- 1$ by $8$ .

$y' = - \frac{\frac{1}{8} v^{\frac{1 - 8}{8}} \frac{d}{d x} [v]}{v^{\frac{1}{4}}}$

Step 5.9.2

Subtract $8$ from $1$ .

$y' = - \frac{\frac{1}{8} v^{\frac{- 7}{8}} \frac{d}{d x} [v]}{v^{\frac{1}{4}}}$

Step 5.10

Move the negative in front of the fraction.

$y' = - \frac{\frac{1}{8} v^{- \frac{7}{8}} \frac{d}{d x} [v]}{v^{\frac{1}{4}}}$

Step 5.11

Combine $\frac{1}{8}$ and $v^{- \frac{7}{8}}$ .

$y' = - \frac{\frac{v^{- \frac{7}{8}}}{8} \frac{d}{d x} [v]}{v^{\frac{1}{4}}}$

Step 5.12

Move $v^{- \frac{7}{8}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y' = - \frac{\frac{1}{8 v^{\frac{7}{8}}} \frac{d}{d x} [v]}{v^{\frac{1}{4}}}$

Step 5.13

Rewrite $\frac{d}{d x} [v]$ as $v'$ .

$y' = - \frac{\frac{1}{8 v^{\frac{7}{8}}} v'}{v^{\frac{1}{4}}}$

Step 5.14

Combine $\frac{1}{8 v^{\frac{7}{8}}}$ and $v'$ .

$y' = - \frac{\frac{v'}{8 v^{\frac{7}{8}}}}{v^{\frac{1}{4}}}$

Step 5.15

Rewrite $\frac{\frac{v'}{8 v^{\frac{7}{8}}}}{v^{\frac{1}{4}}}$ as a product.

$y' = - (\frac{v'}{8 v^{\frac{7}{8}}} \cdot \frac{1}{v^{\frac{1}{4}}})$

Step 5.16

Multiply $\frac{v'}{8 v^{\frac{7}{8}}}$ by $\frac{1}{v^{\frac{1}{4}}}$ .

$y' = - \frac{v'}{8 v^{\frac{7}{8}} v^{\frac{1}{4}}}$

Step 5.17

Multiply $v^{\frac{7}{8}}$ by $v^{\frac{1}{4}}$ by adding the exponents.

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Step 5.17.1

Move $v^{\frac{1}{4}}$ .

$y' = - \frac{v'}{8 (v^{\frac{1}{4}} v^{\frac{7}{8}})}$

Step 5.17.2

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

$y' = - \frac{v'}{8 v^{\frac{1}{4} + \frac{7}{8}}}$

Step 5.17.3

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y' = - \frac{v'}{8 v^{\frac{1}{4} \cdot \frac{2}{2} + \frac{7}{8}}}$

Step 5.17.4

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.17.4.1

Multiply $\frac{1}{4}$ by $\frac{2}{2}$ .

$y' = - \frac{v'}{8 v^{\frac{2}{4 \cdot 2} + \frac{7}{8}}}$

Step 5.17.4.2

Multiply $4$ by $2$ .

$y' = - \frac{v'}{8 v^{\frac{2}{8} + \frac{7}{8}}}$

Step 5.17.5

Combine the numerators over the common denominator.

$y' = - \frac{v'}{8 v^{\frac{2 + 7}{8}}}$

Step 5.17.6

Add $2$ and $7$ .

$y' = - \frac{v'}{8 v^{\frac{9}{8}}}$

Step 6

Substitute $- \frac{v'}{8 v^{\frac{9}{8}}}$ for $\frac{d y}{d x}$ and $v^{- \frac{1}{8}}$ for $y$ in the original equation $\frac{d y}{d x} + \frac{1}{x} y = y^{9}$ .

$- \frac{v'}{8 v^{\frac{9}{8}}} + \frac{1}{x} v^{- \frac{1}{8}} = {(v^{- \frac{1}{8}})}^{9}$

Step 7

Solve the substituted differential equation.

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Step 7.1

Rewrite the differential equation as $\frac{d v}{d x} + M (x) v = Q (x)$ .

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Step 7.1.1

Multiply each term in $- \frac{\frac{d v}{d x}}{8 v^{\frac{9}{8}}} + \frac{1}{x} v^{- \frac{1}{8}} = {(v^{- \frac{1}{8}})}^{9}$ by $- (8 v^{\frac{9}{8}})$ to eliminate the fractions.

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Step 7.1.1.1

Multiply each term in $- \frac{\frac{d v}{d x}}{8 v^{\frac{9}{8}}} + \frac{1}{x} v^{- \frac{1}{8}} = {(v^{- \frac{1}{8}})}^{9}$ by $- (8 v^{\frac{9}{8}})$ .

$- \frac{\frac{d v}{d x}}{8 v^{\frac{9}{8}}} (- (8 v^{\frac{9}{8}})) + \frac{1}{x} v^{- \frac{1}{8}} (- (8 v^{\frac{9}{8}})) = {(v^{- \frac{1}{8}})}^{9} (- (8 v^{\frac{9}{8}}))$

Step 7.1.1.2

Simplify the left side.

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Step 7.1.1.2.1

Simplify each term.

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Step 7.1.1.2.1.1

Cancel the common factor of $8 v^{\frac{9}{8}}$ .

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Step 7.1.1.2.1.1.1

Move the leading negative in $- \frac{\frac{d v}{d x}}{8 v^{\frac{9}{8}}}$ into the numerator.

$\frac{- \frac{d v}{d x}}{8 v^{\frac{9}{8}}} (- (8 v^{\frac{9}{8}})) + \frac{1}{x} v^{- \frac{1}{8}} (- (8 v^{\frac{9}{8}})) = {(v^{- \frac{1}{8}})}^{9} (- (8 v^{\frac{9}{8}}))$

Step 7.1.1.2.1.1.2

Factor $8 v^{\frac{9}{8}}$ out of $- (8 v^{\frac{9}{8}})$ .

$\frac{- \frac{d v}{d x}}{8 v^{\frac{9}{8}}} (8 v^{\frac{9}{8}} (- 1)) + \frac{1}{x} v^{- \frac{1}{8}} (- (8 v^{\frac{9}{8}})) = {(v^{- \frac{1}{8}})}^{9} (- (8 v^{\frac{9}{8}}))$

Step 7.1.1.2.1.1.3

Cancel the common factor.

$\frac{- \frac{d v}{d x}}{8 v^{\frac{9}{8}}} (8 v^{\frac{9}{8}} \cdot - 1) + \frac{1}{x} v^{- \frac{1}{8}} (- (8 v^{\frac{9}{8}})) = {(v^{- \frac{1}{8}})}^{9} (- (8 v^{\frac{9}{8}}))$

Step 7.1.1.2.1.1.4

Rewrite the expression.

$- \frac{d v}{d x} \cdot - 1 + \frac{1}{x} v^{- \frac{1}{8}} (- (8 v^{\frac{9}{8}})) = {(v^{- \frac{1}{8}})}^{9} (- (8 v^{\frac{9}{8}}))$

Step 7.1.1.2.1.2

Multiply $- 1$ by $- 1$ .

$1 \frac{d v}{d x} + \frac{1}{x} v^{- \frac{1}{8}} (- (8 v^{\frac{9}{8}})) = {(v^{- \frac{1}{8}})}^{9} (- (8 v^{\frac{9}{8}}))$

Step 7.1.1.2.1.3

Multiply $\frac{d v}{d x}$ by $1$ .

$\frac{d v}{d x} + \frac{1}{x} v^{- \frac{1}{8}} (- (8 v^{\frac{9}{8}})) = {(v^{- \frac{1}{8}})}^{9} (- (8 v^{\frac{9}{8}}))$

Step 7.1.1.2.1.4

Rewrite using the commutative property of multiplication.

$\frac{d v}{d x} - 1 \cdot 8 \frac{1}{x} v^{- \frac{1}{8}} v^{\frac{9}{8}} = {(v^{- \frac{1}{8}})}^{9} (- (8 v^{\frac{9}{8}}))$

Step 7.1.1.2.1.5

Multiply $v^{- \frac{1}{8}}$ by $v^{\frac{9}{8}}$ by adding the exponents.

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Step 7.1.1.2.1.5.1

Move $v^{\frac{9}{8}}$ .

$\frac{d v}{d x} - 1 \cdot 8 \frac{1}{x} (v^{\frac{9}{8}} v^{- \frac{1}{8}}) = {(v^{- \frac{1}{8}})}^{9} (- (8 v^{\frac{9}{8}}))$

Step 7.1.1.2.1.5.2

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

$\frac{d v}{d x} - 1 \cdot 8 \frac{1}{x} v^{\frac{9}{8} - \frac{1}{8}} = {(v^{- \frac{1}{8}})}^{9} (- (8 v^{\frac{9}{8}}))$

Step 7.1.1.2.1.5.3

Combine the numerators over the common denominator.

$\frac{d v}{d x} - 1 \cdot 8 \frac{1}{x} v^{\frac{9 - 1}{8}} = {(v^{- \frac{1}{8}})}^{9} (- (8 v^{\frac{9}{8}}))$

Step 7.1.1.2.1.5.4

Subtract $1$ from $9$ .

$\frac{d v}{d x} - 1 \cdot 8 \frac{1}{x} v^{\frac{8}{8}} = {(v^{- \frac{1}{8}})}^{9} (- (8 v^{\frac{9}{8}}))$

Step 7.1.1.2.1.5.5

Divide $8$ by $8$ .

$\frac{d v}{d x} - 1 \cdot 8 \frac{1}{x} v^{1} = {(v^{- \frac{1}{8}})}^{9} (- (8 v^{\frac{9}{8}}))$

Step 7.1.1.2.1.6

Simplify $- 1 \cdot 8 \frac{1}{x} v^{1}$ .

$\frac{d v}{d x} - 1 \cdot 8 \frac{1}{x} v = {(v^{- \frac{1}{8}})}^{9} (- (8 v^{\frac{9}{8}}))$

Step 7.1.1.2.1.7

Multiply $- 1$ by $8$ .

$\frac{d v}{d x} - 8 \frac{1}{x} v = {(v^{- \frac{1}{8}})}^{9} (- (8 v^{\frac{9}{8}}))$

Step 7.1.1.2.1.8

Combine $- 8$ and $\frac{1}{x}$ .

$\frac{d v}{d x} + \frac{- 8}{x} v = {(v^{- \frac{1}{8}})}^{9} (- (8 v^{\frac{9}{8}}))$

Step 7.1.1.2.1.9

Move the negative in front of the fraction.

$\frac{d v}{d x} - \frac{8}{x} v = {(v^{- \frac{1}{8}})}^{9} (- (8 v^{\frac{9}{8}}))$

Step 7.1.1.2.1.10

Combine $v$ and $\frac{8}{x}$ .

$\frac{d v}{d x} - \frac{v \cdot 8}{x} = {(v^{- \frac{1}{8}})}^{9} (- (8 v^{\frac{9}{8}}))$

Step 7.1.1.2.1.11

Move $8$ to the left of $v$ .

$\frac{d v}{d x} - \frac{8 v}{x} = {(v^{- \frac{1}{8}})}^{9} (- (8 v^{\frac{9}{8}}))$

Step 7.1.1.3

Simplify the right side.

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Step 7.1.1.3.1

Rewrite using the commutative property of multiplication.

$\frac{d v}{d x} - \frac{8 v}{x} = - 1 \cdot 8 {(v^{- \frac{1}{8}})}^{9} v^{\frac{9}{8}}$

Step 7.1.1.3.2

Multiply $- 1$ by $8$ .

$\frac{d v}{d x} - \frac{8 v}{x} = - 8 {(v^{- \frac{1}{8}})}^{9} v^{\frac{9}{8}}$

Step 7.1.1.3.3

Multiply the exponents in ${(v^{- \frac{1}{8}})}^{9}$ .

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Step 7.1.1.3.3.1

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

$\frac{d v}{d x} - \frac{8 v}{x} = - 8 v^{- \frac{1}{8} \cdot 9} v^{\frac{9}{8}}$

Step 7.1.1.3.3.2

Multiply $- \frac{1}{8} \cdot 9$ .

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Step 7.1.1.3.3.2.1

Multiply $9$ by $- 1$ .

$\frac{d v}{d x} - \frac{8 v}{x} = - 8 v^{- 9 (\frac{1}{8})} v^{\frac{9}{8}}$

Step 7.1.1.3.3.2.2

Combine $- 9$ and $\frac{1}{8}$ .

$\frac{d v}{d x} - \frac{8 v}{x} = - 8 v^{\frac{- 9}{8}} v^{\frac{9}{8}}$

Step 7.1.1.3.3.3

Move the negative in front of the fraction.

$\frac{d v}{d x} - \frac{8 v}{x} = - 8 v^{- \frac{9}{8}} v^{\frac{9}{8}}$

Step 7.1.1.3.4

Multiply $v^{- \frac{9}{8}}$ by $v^{\frac{9}{8}}$ by adding the exponents.

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Step 7.1.1.3.4.1

Move $v^{\frac{9}{8}}$ .

$\frac{d v}{d x} - \frac{8 v}{x} = - 8 (v^{\frac{9}{8}} v^{- \frac{9}{8}})$

Step 7.1.1.3.4.2

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

$\frac{d v}{d x} - \frac{8 v}{x} = - 8 v^{\frac{9}{8} - \frac{9}{8}}$

Step 7.1.1.3.4.3

Combine the numerators over the common denominator.

$\frac{d v}{d x} - \frac{8 v}{x} = - 8 v^{\frac{9 - 9}{8}}$

Step 7.1.1.3.4.4

Subtract $9$ from $9$ .

$\frac{d v}{d x} - \frac{8 v}{x} = - 8 v^{\frac{0}{8}}$

Step 7.1.1.3.4.5

Divide $0$ by $8$ .

$\frac{d v}{d x} - \frac{8 v}{x} = - 8 v^{0}$

Step 7.1.1.3.5

Simplify $- 8 v^{0}$ .

$\frac{d v}{d x} - \frac{8 v}{x} = - 8$

Step 7.1.2

Factor $v$ out of $- \frac{8 v}{x}$ .

$\frac{d v}{d x} + v (- \frac{8}{x}) = - 8$

Step 7.1.3

Reorder $v$ and $- \frac{8}{x}$ .

$\frac{d v}{d x} - \frac{8}{x} v = - 8$

Step 7.2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = - \frac{8}{x}$ .

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Step 7.2.1

Set up the integration.

$e^{\int - \frac{8}{x} d x}$

Step 7.2.2

Integrate $- \frac{8}{x}$ .

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Step 7.2.2.1

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

$e^{- \int \frac{8}{x} d x}$

Step 7.2.2.2

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$e^{- (8 \int \frac{1}{x} d x)}$

Step 7.2.2.3

Multiply $8$ by $- 1$ .

$e^{- 8 \int \frac{1}{x} d x}$

Step 7.2.2.4

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

$e^{- 8 (\ln (| x |) + C)}$

Step 7.2.2.5

Simplify.

$e^{- 8 \ln (| x |) + C}$

Step 7.2.3

Remove the constant of integration.

$e^{- 8 \ln (x)}$

Step 7.2.4

Use the logarithmic power rule.

$e^{\ln (x^{- 8})}$

Step 7.2.5

Exponentiation and log are inverse functions.

$x^{- 8}$

Step 7.2.6

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

$\frac{1}{x^{8}}$

Step 7.3

Multiply each term by the integrating factor $\frac{1}{x^{8}}$ .

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Step 7.3.1

Multiply each term by $\frac{1}{x^{8}}$ .

$\frac{1}{x^{8}} \frac{d v}{d x} + \frac{1}{x^{8}} (- \frac{8}{x} v) = \frac{1}{x^{8}} \cdot - 8$

Step 7.3.2

Simplify each term.

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Step 7.3.2.1

Combine $\frac{1}{x^{8}}$ and $\frac{d v}{d x}$ .

$\frac{\frac{d v}{d x}}{x^{8}} + \frac{1}{x^{8}} (- \frac{8}{x} v) = \frac{1}{x^{8}} \cdot - 8$

Step 7.3.2.2

Rewrite using the commutative property of multiplication.

$\frac{\frac{d v}{d x}}{x^{8}} - \frac{1}{x^{8}} (\frac{8}{x} v) = \frac{1}{x^{8}} \cdot - 8$

Step 7.3.2.3

Combine $\frac{8}{x}$ and $v$ .

$\frac{\frac{d v}{d x}}{x^{8}} - \frac{1}{x^{8}} \cdot \frac{8 v}{x} = \frac{1}{x^{8}} \cdot - 8$

Step 7.3.2.4

Multiply $- \frac{1}{x^{8}} \cdot \frac{8 v}{x}$ .

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Step 7.3.2.4.1

Multiply $\frac{8 v}{x}$ by $\frac{1}{x^{8}}$ .

$\frac{\frac{d v}{d x}}{x^{8}} - \frac{8 v}{x \cdot x^{8}} = \frac{1}{x^{8}} \cdot - 8$

Step 7.3.2.4.2

Multiply $x$ by $x^{8}$ by adding the exponents.

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Step 7.3.2.4.2.1

Multiply $x$ by $x^{8}$ .

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Step 7.3.2.4.2.1.1

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

$\frac{\frac{d v}{d x}}{x^{8}} - \frac{8 v}{x^{1} x^{8}} = \frac{1}{x^{8}} \cdot - 8$

Step 7.3.2.4.2.1.2

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

$\frac{\frac{d v}{d x}}{x^{8}} - \frac{8 v}{x^{1 + 8}} = \frac{1}{x^{8}} \cdot - 8$

Step 7.3.2.4.2.2

Add $1$ and $8$ .

$\frac{\frac{d v}{d x}}{x^{8}} - \frac{8 v}{x^{9}} = \frac{1}{x^{8}} \cdot - 8$

Step 7.3.3

Combine $\frac{1}{x^{8}}$ and $- 8$ .

$\frac{\frac{d v}{d x}}{x^{8}} - \frac{8 v}{x^{9}} = \frac{- 8}{x^{8}}$

Step 7.3.4

Move the negative in front of the fraction.

$\frac{\frac{d v}{d x}}{x^{8}} - \frac{8 v}{x^{9}} = - \frac{8}{x^{8}}$

Step 7.4

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

$\frac{d}{d x} [\frac{1}{x^{8}} v] = - \frac{8}{x^{8}}$

Step 7.5

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x^{8}} v] d x = \int - \frac{8}{x^{8}} d x$

Step 7.6

Integrate the left side.

$\frac{1}{x^{8}} v = \int - \frac{8}{x^{8}} d x$

Step 7.7

Integrate the right side.

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Step 7.7.1

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

$\frac{1}{x^{8}} v = - \int \frac{8}{x^{8}} d x$

Step 7.7.2

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$\frac{1}{x^{8}} v = - (8 \int \frac{1}{x^{8}} d x)$

Step 7.7.3

Simplify the expression.

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Step 7.7.3.1

Multiply $8$ by $- 1$ .

$\frac{1}{x^{8}} v = - 8 \int \frac{1}{x^{8}} d x$

Step 7.7.3.2

Move $x^{8}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{x^{8}} v = - 8 \int {(x^{8})}^{- 1} d x$

Step 7.7.3.3

Multiply the exponents in ${(x^{8})}^{- 1}$ .

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Step 7.7.3.3.1

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

$\frac{1}{x^{8}} v = - 8 \int x^{8 \cdot - 1} d x$

Step 7.7.3.3.2

Multiply $8$ by $- 1$ .

$\frac{1}{x^{8}} v = - 8 \int x^{- 8} d x$

Step 7.7.4

By the Power Rule, the integral of $x^{- 8}$ with respect to $x$ is $- \frac{1}{7} x^{- 7}$ .

$\frac{1}{x^{8}} v = - 8 (- \frac{1}{7} x^{- 7} + C)$

Step 7.7.5

Simplify the answer.

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Step 7.7.5.1

Simplify.

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Step 7.7.5.1.1

Combine $x^{- 7}$ and $\frac{1}{7}$ .

$\frac{1}{x^{8}} v = - 8 (- \frac{x^{- 7}}{7} + C)$

Step 7.7.5.1.2

Move $x^{- 7}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{8}} v = - 8 (- \frac{1}{7 x^{7}} + C)$

Step 7.7.5.2

Simplify.

$\frac{1}{x^{8}} v = - 8 (- \frac{1}{7 x^{7}}) + C$

Step 7.7.5.3

Simplify.

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Step 7.7.5.3.1

Multiply $- 1$ by $- 8$ .

$\frac{1}{x^{8}} v = 8 \frac{1}{7 x^{7}} + C$

Step 7.7.5.3.2

Combine $8$ and $\frac{1}{7 x^{7}}$ .

$\frac{1}{x^{8}} v = \frac{8}{7 x^{7}} + C$

Step 7.8

Solve for $v$ .

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Step 7.8.1

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

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Step 7.8.1.1

Subtract $\frac{8}{7 x^{7}}$ from both sides of the equation.

$\frac{1}{x^{8}} v - \frac{8}{7 x^{7}} = C$

Step 7.8.1.2

Subtract $C$ from both sides of the equation.

$\frac{1}{x^{8}} v - \frac{8}{7 x^{7}} - C = 0$

Step 7.8.1.3

Combine $\frac{1}{x^{8}}$ and $v$ .

$\frac{v}{x^{8}} - \frac{8}{7 x^{7}} - C = 0$

Step 7.8.2

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

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Step 7.8.2.1

Add $\frac{8}{7 x^{7}}$ to both sides of the equation.

$\frac{v}{x^{8}} - C = \frac{8}{7 x^{7}}$

Step 7.8.2.2

Add $C$ to both sides of the equation.

$\frac{v}{x^{8}} = \frac{8}{7 x^{7}} + C$

Step 7.8.3

Multiply both sides by $x^{8}$ .

$\frac{v}{x^{8}} x^{8} = (\frac{8}{7 x^{7}} + C) x^{8}$

Step 7.8.4

Simplify.

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Step 7.8.4.1

Simplify the left side.

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Step 7.8.4.1.1

Cancel the common factor of $x^{8}$ .

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Step 7.8.4.1.1.1

Cancel the common factor.

$\frac{v}{x^{8}} x^{8} = (\frac{8}{7 x^{7}} + C) x^{8}$

Step 7.8.4.1.1.2

Rewrite the expression.

$v = (\frac{8}{7 x^{7}} + C) x^{8}$

Step 7.8.4.2

Simplify the right side.

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Step 7.8.4.2.1

Simplify $(\frac{8}{7 x^{7}} + C) x^{8}$ .

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Step 7.8.4.2.1.1

Apply the distributive property.

$v = \frac{8}{7 x^{7}} x^{8} + C x^{8}$

Step 7.8.4.2.1.2

Cancel the common factor of $x^{7}$ .

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Step 7.8.4.2.1.2.1

Factor $x^{7}$ out of $7 x^{7}$ .

$v = \frac{8}{x^{7} \cdot 7} x^{8} + C x^{8}$

Step 7.8.4.2.1.2.2

Factor $x^{7}$ out of $x^{8}$ .

$v = \frac{8}{x^{7} \cdot 7} (x^{7} x) + C x^{8}$

Step 7.8.4.2.1.2.3

Cancel the common factor.

$v = \frac{8}{x^{7} \cdot 7} (x^{7} x) + C x^{8}$

Step 7.8.4.2.1.2.4

Rewrite the expression.

$v = \frac{8}{7} x + C x^{8}$

Step 7.8.4.2.1.3

Combine $\frac{8}{7}$ and $x$ .

$v = \frac{8 x}{7} + C x^{8}$

Step 7.8.4.2.1.4

Reorder $\frac{8 x}{7}$ and $C x^{8}$ .

$v = C x^{8} + \frac{8 x}{7}$

Step 8

Substitute $y^{- 8}$ for $v$ .

$y^{- 8} = C x^{8} + \frac{8 x}{7}$