Calculus Examples

$\frac{d y}{d x} - \frac{2}{x} y = \frac{3}{x^{2}} y^{4}$

Step 1

To solve the differential equation, let $v = y^{1 - n}$ where $n$ is the exponent of $y^{4}$ .

$v = y^{- 3}$

Step 2

Solve the equation for $y$ .

$y = v^{- \frac{1}{3}}$

Step 3

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

$y' = v^{- \frac{1}{3}}$

Step 4

Take the derivative of $v^{- \frac{1}{3}}$ with respect to $x$ .

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Step 4.1

Take the derivative of $v^{- \frac{1}{3}}$ .

$y' = \frac{d}{d x} [v^{- \frac{1}{3}}]$

Step 4.2

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

$y' = \frac{d}{d x} [\frac{1}{v^{\frac{1}{3}}}]$

Step 4.3

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 $g (x) = v^{\frac{1}{3}}$ .

$y' = \frac{v^{\frac{1}{3}} \frac{d}{d x} [1] - 1 \cdot 1 \frac{d}{d x} [v^{\frac{1}{3}}]}{{(v^{\frac{1}{3}})}^{2}}$

Step 4.4

Differentiate using the Constant Rule.

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Step 4.4.1

Multiply $- 1$ by $1$ .

$y' = \frac{v^{\frac{1}{3}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{3}}]}{{(v^{\frac{1}{3}})}^{2}}$

Step 4.4.2

Multiply the exponents in ${(v^{\frac{1}{3}})}^{2}$ .

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Step 4.4.2.1

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

$y' = \frac{v^{\frac{1}{3}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{3}}]}{v^{\frac{1}{3} \cdot 2}}$

Step 4.4.2.2

Combine $\frac{1}{3}$ and $2$ .

$y' = \frac{v^{\frac{1}{3}} \frac{d}{d x} [1] - \frac{d}{d x} [v^{\frac{1}{3}}]}{v^{\frac{2}{3}}}$

Step 4.4.3

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

$y' = \frac{v^{\frac{1}{3}} \cdot 0 - \frac{d}{d x} [v^{\frac{1}{3}}]}{v^{\frac{2}{3}}}$

Step 4.4.4

Simplify the expression.

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Step 4.4.4.1

Multiply $v^{\frac{1}{3}}$ by $0$ .

$y' = \frac{0 - \frac{d}{d x} [v^{\frac{1}{3}}]}{v^{\frac{2}{3}}}$

Step 4.4.4.2

Subtract $\frac{d}{d x} [v^{\frac{1}{3}}]$ from $0$ .

$y' = \frac{- \frac{d}{d x} [v^{\frac{1}{3}}]}{v^{\frac{2}{3}}}$

Step 4.4.4.3

Move the negative in front of the fraction.

$y' = - \frac{\frac{d}{d x} [v^{\frac{1}{3}}]}{v^{\frac{2}{3}}}$

Step 4.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = v$ .

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Step 4.5.1

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

$y' = - \frac{\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [v]}{v^{\frac{2}{3}}}$

Step 4.5.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{3}$ .

$y' = - \frac{\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [v]}{v^{\frac{2}{3}}}$

Step 4.5.3

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

$y' = - \frac{\frac{1}{3} v^{\frac{1}{3} - 1} \frac{d}{d x} [v]}{v^{\frac{2}{3}}}$

Step 4.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y' = - \frac{\frac{1}{3} v^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [v]}{v^{\frac{2}{3}}}$

Step 4.7

Combine $- 1$ and $\frac{3}{3}$ .

$y' = - \frac{\frac{1}{3} v^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [v]}{v^{\frac{2}{3}}}$

Step 4.8

Combine the numerators over the common denominator.

$y' = - \frac{\frac{1}{3} v^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} [v]}{v^{\frac{2}{3}}}$

Step 4.9

Simplify the numerator.

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Step 4.9.1

Multiply $- 1$ by $3$ .

$y' = - \frac{\frac{1}{3} v^{\frac{1 - 3}{3}} \frac{d}{d x} [v]}{v^{\frac{2}{3}}}$

Step 4.9.2

Subtract $3$ from $1$ .

$y' = - \frac{\frac{1}{3} v^{\frac{- 2}{3}} \frac{d}{d x} [v]}{v^{\frac{2}{3}}}$

Step 4.10

Move the negative in front of the fraction.

$y' = - \frac{\frac{1}{3} v^{- \frac{2}{3}} \frac{d}{d x} [v]}{v^{\frac{2}{3}}}$

Step 4.11

Combine $\frac{1}{3}$ and $v^{- \frac{2}{3}}$ .

$y' = - \frac{\frac{v^{- \frac{2}{3}}}{3} \frac{d}{d x} [v]}{v^{\frac{2}{3}}}$

Step 4.12

Move $v^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y' = - \frac{\frac{1}{3 v^{\frac{2}{3}}} \frac{d}{d x} [v]}{v^{\frac{2}{3}}}$

Step 4.13

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

$y' = - \frac{\frac{1}{3 v^{\frac{2}{3}}} v'}{v^{\frac{2}{3}}}$

Step 4.14

Combine $\frac{1}{3 v^{\frac{2}{3}}}$ and $v'$ .

$y' = - \frac{\frac{v'}{3 v^{\frac{2}{3}}}}{v^{\frac{2}{3}}}$

Step 4.15

Rewrite $\frac{\frac{v'}{3 v^{\frac{2}{3}}}}{v^{\frac{2}{3}}}$ as a product.

$y' = - (\frac{v'}{3 v^{\frac{2}{3}}} \cdot \frac{1}{v^{\frac{2}{3}}})$

Step 4.16

Multiply $\frac{v'}{3 v^{\frac{2}{3}}}$ by $\frac{1}{v^{\frac{2}{3}}}$ .

$y' = - \frac{v'}{3 v^{\frac{2}{3}} v^{\frac{2}{3}}}$

Step 4.17

Multiply $v^{\frac{2}{3}}$ by $v^{\frac{2}{3}}$ by adding the exponents.

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Step 4.17.1

Move $v^{\frac{2}{3}}$ .

$y' = - \frac{v'}{3 (v^{\frac{2}{3}} v^{\frac{2}{3}})}$

Step 4.17.2

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

$y' = - \frac{v'}{3 v^{\frac{2}{3} + \frac{2}{3}}}$

Step 4.17.3

Combine the numerators over the common denominator.

$y' = - \frac{v'}{3 v^{\frac{2 + 2}{3}}}$

Step 4.17.4

Add $2$ and $2$ .

$y' = - \frac{v'}{3 v^{\frac{4}{3}}}$

Step 5

Substitute $- \frac{v'}{3 v^{\frac{4}{3}}}$ for $\frac{d y}{d x}$ and $v^{- \frac{1}{3}}$ for $y$ in the original equation $\frac{d y}{d x} - \frac{2}{x} y = \frac{3}{x^{2}} y^{4}$ .

$- \frac{v'}{3 v^{\frac{4}{3}}} - \frac{2}{x} v^{- \frac{1}{3}} = \frac{3}{x^{2}} {(v^{- \frac{1}{3}})}^{4}$

Step 6

Solve the substituted differential equation.

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Step 6.1

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

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Step 6.1.1

Multiply each term in $- \frac{\frac{d v}{d x}}{3 v^{\frac{4}{3}}} - \frac{2}{x} v^{- \frac{1}{3}} = \frac{3}{x^{2}} {(v^{- \frac{1}{3}})}^{4}$ by $- (3 v^{\frac{4}{3}})$ to eliminate the fractions.

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Step 6.1.1.1

Multiply each term in $- \frac{\frac{d v}{d x}}{3 v^{\frac{4}{3}}} - \frac{2}{x} v^{- \frac{1}{3}} = \frac{3}{x^{2}} {(v^{- \frac{1}{3}})}^{4}$ by $- (3 v^{\frac{4}{3}})$ .

$- \frac{\frac{d v}{d x}}{3 v^{\frac{4}{3}}} (- (3 v^{\frac{4}{3}})) - \frac{2}{x} v^{- \frac{1}{3}} (- (3 v^{\frac{4}{3}})) = \frac{3}{x^{2}} {(v^{- \frac{1}{3}})}^{4} (- (3 v^{\frac{4}{3}}))$

Step 6.1.1.2

Simplify the left side.

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Step 6.1.1.2.1

Simplify each term.

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Step 6.1.1.2.1.1

Cancel the common factor of $3 v^{\frac{4}{3}}$ .

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Step 6.1.1.2.1.1.1

Move the leading negative in $- \frac{\frac{d v}{d x}}{3 v^{\frac{4}{3}}}$ into the numerator.

$\frac{- \frac{d v}{d x}}{3 v^{\frac{4}{3}}} (- (3 v^{\frac{4}{3}})) - \frac{2}{x} v^{- \frac{1}{3}} (- (3 v^{\frac{4}{3}})) = \frac{3}{x^{2}} {(v^{- \frac{1}{3}})}^{4} (- (3 v^{\frac{4}{3}}))$

Step 6.1.1.2.1.1.2

Factor $3 v^{\frac{4}{3}}$ out of $- (3 v^{\frac{4}{3}})$ .

$\frac{- \frac{d v}{d x}}{3 v^{\frac{4}{3}}} (3 v^{\frac{4}{3}} (- 1)) - \frac{2}{x} v^{- \frac{1}{3}} (- (3 v^{\frac{4}{3}})) = \frac{3}{x^{2}} {(v^{- \frac{1}{3}})}^{4} (- (3 v^{\frac{4}{3}}))$

Step 6.1.1.2.1.1.3

Cancel the common factor.

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

Step 6.1.1.2.1.1.4

Rewrite the expression.

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

Step 6.1.1.2.1.2

Multiply $- 1$ by $- 1$ .

$1 \frac{d v}{d x} - \frac{2}{x} v^{- \frac{1}{3}} (- (3 v^{\frac{4}{3}})) = \frac{3}{x^{2}} {(v^{- \frac{1}{3}})}^{4} (- (3 v^{\frac{4}{3}}))$

Step 6.1.1.2.1.3

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

$\frac{d v}{d x} - \frac{2}{x} v^{- \frac{1}{3}} (- (3 v^{\frac{4}{3}})) = \frac{3}{x^{2}} {(v^{- \frac{1}{3}})}^{4} (- (3 v^{\frac{4}{3}}))$

Step 6.1.1.2.1.4

Multiply $v^{- \frac{1}{3}}$ by $v^{\frac{4}{3}}$ by adding the exponents.

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Step 6.1.1.2.1.4.1

Move $v^{\frac{4}{3}}$ .

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

Step 6.1.1.2.1.4.2

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

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

Step 6.1.1.2.1.4.3

Combine the numerators over the common denominator.

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

Step 6.1.1.2.1.4.4

Subtract $1$ from $4$ .

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

Step 6.1.1.2.1.4.5

Divide $3$ by $3$ .

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

Step 6.1.1.2.1.5

Simplify $- \frac{2}{x} v^{1} (- 1 \cdot (3))$ .

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

Step 6.1.1.2.1.6

Combine $v$ and $\frac{2}{x}$ .

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

Step 6.1.1.2.1.7

Move $2$ to the left of $v$ .

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

Step 6.1.1.2.1.8

Multiply $- 1$ by $3$ .

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

Step 6.1.1.2.1.9

Multiply $- \frac{2 v}{x} \cdot - 3$ .

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Step 6.1.1.2.1.9.1

Multiply $- 3$ by $- 1$ .

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

Step 6.1.1.2.1.9.2

Combine $3$ and $\frac{2 v}{x}$ .

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

Step 6.1.1.2.1.9.3

Multiply $2$ by $3$ .

$\frac{d v}{d x} + \frac{6 v}{x} = \frac{3}{x^{2}} {(v^{- \frac{1}{3}})}^{4} (- (3 v^{\frac{4}{3}}))$

Step 6.1.1.3

Simplify the right side.

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Step 6.1.1.3.1

Rewrite using the commutative property of multiplication.

$\frac{d v}{d x} + \frac{6 v}{x} = - 1 \cdot 3 \frac{3}{x^{2}} {(v^{- \frac{1}{3}})}^{4} v^{\frac{4}{3}}$

Step 6.1.1.3.2

Multiply $- 1$ by $3$ .

$\frac{d v}{d x} + \frac{6 v}{x} = - 3 \frac{3}{x^{2}} {(v^{- \frac{1}{3}})}^{4} v^{\frac{4}{3}}$

Step 6.1.1.3.3

Multiply $- 3 \frac{3}{x^{2}}$ .

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Step 6.1.1.3.3.1

Combine $- 3$ and $\frac{3}{x^{2}}$ .

$\frac{d v}{d x} + \frac{6 v}{x} = \frac{- 3 \cdot 3}{x^{2}} {(v^{- \frac{1}{3}})}^{4} v^{\frac{4}{3}}$

Step 6.1.1.3.3.2

Multiply $- 3$ by $3$ .

$\frac{d v}{d x} + \frac{6 v}{x} = \frac{- 9}{x^{2}} {(v^{- \frac{1}{3}})}^{4} v^{\frac{4}{3}}$

Step 6.1.1.3.4

Move the negative in front of the fraction.

$\frac{d v}{d x} + \frac{6 v}{x} = - \frac{9}{x^{2}} {(v^{- \frac{1}{3}})}^{4} v^{\frac{4}{3}}$

Step 6.1.1.3.5

Multiply the exponents in ${(v^{- \frac{1}{3}})}^{4}$ .

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Step 6.1.1.3.5.1

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

$\frac{d v}{d x} + \frac{6 v}{x} = - \frac{9}{x^{2}} v^{- \frac{1}{3} \cdot 4} v^{\frac{4}{3}}$

Step 6.1.1.3.5.2

Multiply $- \frac{1}{3} \cdot 4$ .

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Step 6.1.1.3.5.2.1

Multiply $4$ by $- 1$ .

$\frac{d v}{d x} + \frac{6 v}{x} = - \frac{9}{x^{2}} v^{- 4 (\frac{1}{3})} v^{\frac{4}{3}}$

Step 6.1.1.3.5.2.2

Combine $- 4$ and $\frac{1}{3}$ .

$\frac{d v}{d x} + \frac{6 v}{x} = - \frac{9}{x^{2}} v^{\frac{- 4}{3}} v^{\frac{4}{3}}$

Step 6.1.1.3.5.3

Move the negative in front of the fraction.

$\frac{d v}{d x} + \frac{6 v}{x} = - \frac{9}{x^{2}} v^{- \frac{4}{3}} v^{\frac{4}{3}}$

Step 6.1.1.3.6

Multiply $v^{- \frac{4}{3}}$ by $v^{\frac{4}{3}}$ by adding the exponents.

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Step 6.1.1.3.6.1

Move $v^{\frac{4}{3}}$ .

$\frac{d v}{d x} + \frac{6 v}{x} = - \frac{9}{x^{2}} (v^{\frac{4}{3}} v^{- \frac{4}{3}})$

Step 6.1.1.3.6.2

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

$\frac{d v}{d x} + \frac{6 v}{x} = - \frac{9}{x^{2}} v^{\frac{4}{3} - \frac{4}{3}}$

Step 6.1.1.3.6.3

Combine the numerators over the common denominator.

$\frac{d v}{d x} + \frac{6 v}{x} = - \frac{9}{x^{2}} v^{\frac{4 - 4}{3}}$

Step 6.1.1.3.6.4

Subtract $4$ from $4$ .

$\frac{d v}{d x} + \frac{6 v}{x} = - \frac{9}{x^{2}} v^{\frac{0}{3}}$

Step 6.1.1.3.6.5

Divide $0$ by $3$ .

$\frac{d v}{d x} + \frac{6 v}{x} = - \frac{9}{x^{2}} v^{0}$

Step 6.1.1.3.7

Simplify $- \frac{9}{x^{2}} v^{0}$ .

$\frac{d v}{d x} + \frac{6 v}{x} = - \frac{9}{x^{2}}$

Step 6.1.2

Factor $v$ out of $\frac{6 v}{x}$ .

$\frac{d v}{d x} + v \frac{6}{x} = - \frac{9}{x^{2}}$

Step 6.1.3

Reorder $v$ and $\frac{6}{x}$ .

$\frac{d v}{d x} + \frac{6}{x} v = - \frac{9}{x^{2}}$

Step 6.2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{6}{x}$ .

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Step 6.2.1

Set up the integration.

$e^{\int \frac{6}{x} d x}$

Step 6.2.2

Integrate $\frac{6}{x}$ .

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Step 6.2.2.1

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$e^{6 \int \frac{1}{x} d x}$

Step 6.2.2.2

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

$e^{6 (\ln (| x |) + C)}$

Step 6.2.2.3

Simplify.

$e^{6 \ln (| x |) + C}$

Step 6.2.3

Remove the constant of integration.

$e^{6 \ln (x)}$

Step 6.2.4

Use the logarithmic power rule.

$e^{\ln (x^{6})}$

Step 6.2.5

Exponentiation and log are inverse functions.

$x^{6}$

Step 6.3

Multiply each term by the integrating factor $x^{6}$ .

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Step 6.3.1

Multiply each term by $x^{6}$ .

$x^{6} \frac{d v}{d x} + x^{6} (\frac{6}{x} v) = x^{6} (- \frac{9}{x^{2}})$

Step 6.3.2

Simplify each term.

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Step 6.3.2.1

Combine $\frac{6}{x}$ and $v$ .

$x^{6} \frac{d v}{d x} + x^{6} \frac{6 v}{x} = x^{6} (- \frac{9}{x^{2}})$

Step 6.3.2.2

Cancel the common factor of $x$ .

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Step 6.3.2.2.1

Factor $x$ out of $x^{6}$ .

$x^{6} \frac{d v}{d x} + x \cdot x^{5} \frac{6 v}{x} = x^{6} (- \frac{9}{x^{2}})$

Step 6.3.2.2.2

Cancel the common factor.

$x^{6} \frac{d v}{d x} + x \cdot x^{5} \frac{6 v}{x} = x^{6} (- \frac{9}{x^{2}})$

Step 6.3.2.2.3

Rewrite the expression.

$x^{6} \frac{d v}{d x} + x^{5} (6 v) = x^{6} (- \frac{9}{x^{2}})$

Step 6.3.2.3

Rewrite using the commutative property of multiplication.

$x^{6} \frac{d v}{d x} + 6 x^{5} v = x^{6} (- \frac{9}{x^{2}})$

Step 6.3.3

Rewrite using the commutative property of multiplication.

$x^{6} \frac{d v}{d x} + 6 x^{5} v = - x^{6} \frac{9}{x^{2}}$

Step 6.3.4

Cancel the common factor of $x^{2}$ .

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Step 6.3.4.1

Factor $x^{2}$ out of $- x^{6}$ .

$x^{6} \frac{d v}{d x} + 6 x^{5} v = x^{2} (- x^{4}) \frac{9}{x^{2}}$

Step 6.3.4.2

Cancel the common factor.

$x^{6} \frac{d v}{d x} + 6 x^{5} v = x^{2} (- x^{4}) \frac{9}{x^{2}}$

Step 6.3.4.3

Rewrite the expression.

$x^{6} \frac{d v}{d x} + 6 x^{5} v = - x^{4} \cdot 9$

Step 6.3.5

Multiply $9$ by $- 1$ .

$x^{6} \frac{d v}{d x} + 6 x^{5} v = - 9 x^{4}$

Step 6.4

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

$\frac{d}{d x} [x^{6} v] = - 9 x^{4}$

Step 6.5

Set up an integral on each side.

$\int \frac{d}{d x} [x^{6} v] d x = \int - 9 x^{4} d x$

Step 6.6

Integrate the left side.

$x^{6} v = \int - 9 x^{4} d x$

Step 6.7

Integrate the right side.

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Step 6.7.1

Since $- 9$ is constant with respect to $x$ , move $- 9$ out of the integral.

$x^{6} v = - 9 \int x^{4} d x$

Step 6.7.2

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$x^{6} v = - 9 (\frac{1}{5} x^{5} + C)$

Step 6.7.3

Simplify the answer.

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Step 6.7.3.1

Rewrite $- 9 (\frac{1}{5} x^{5} + C)$ as $- 9 (\frac{1}{5}) x^{5} + C$ .

$x^{6} v = - 9 (\frac{1}{5}) x^{5} + C$

Step 6.7.3.2

Simplify.

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Step 6.7.3.2.1

Combine $- 9$ and $\frac{1}{5}$ .

$x^{6} v = \frac{- 9}{5} x^{5} + C$

Step 6.7.3.2.2

Move the negative in front of the fraction.

$x^{6} v = - \frac{9}{5} x^{5} + C$

Step 6.8

Divide each term in $x^{6} v = - \frac{9}{5} x^{5} + C$ by $x^{6}$ and simplify.

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Step 6.8.1

Divide each term in $x^{6} v = - \frac{9}{5} x^{5} + C$ by $x^{6}$ .

$\frac{x^{6} v}{x^{6}} = \frac{- \frac{9}{5} x^{5}}{x^{6}} + \frac{C}{x^{6}}$

Step 6.8.2

Simplify the left side.

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Step 6.8.2.1

Cancel the common factor of $x^{6}$ .

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Step 6.8.2.1.1

Cancel the common factor.

$\frac{x^{6} v}{x^{6}} = \frac{- \frac{9}{5} x^{5}}{x^{6}} + \frac{C}{x^{6}}$

Step 6.8.2.1.2

Divide $v$ by $1$ .

$v = \frac{- \frac{9}{5} x^{5}}{x^{6}} + \frac{C}{x^{6}}$

Step 6.8.3

Simplify the right side.

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Step 6.8.3.1

Simplify each term.

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Step 6.8.3.1.1

Cancel the common factor of $x^{5}$ and $x^{6}$ .

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Step 6.8.3.1.1.1

Factor $x^{5}$ out of $- \frac{9}{5} x^{5}$ .

$v = \frac{x^{5} (- \frac{9}{5})}{x^{6}} + \frac{C}{x^{6}}$

Step 6.8.3.1.1.2

Cancel the common factors.

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Step 6.8.3.1.1.2.1

Factor $x^{5}$ out of $x^{6}$ .

$v = \frac{x^{5} (- \frac{9}{5})}{x^{5} x} + \frac{C}{x^{6}}$

Step 6.8.3.1.1.2.2

Cancel the common factor.

$v = \frac{x^{5} (- \frac{9}{5})}{x^{5} x} + \frac{C}{x^{6}}$

Step 6.8.3.1.1.2.3

Rewrite the expression.

$v = \frac{- \frac{9}{5}}{x} + \frac{C}{x^{6}}$

Step 6.8.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$v = - \frac{9}{5} \cdot \frac{1}{x} + \frac{C}{x^{6}}$

Step 6.8.3.1.3

Multiply $\frac{1}{x}$ by $\frac{9}{5}$ .

$v = - \frac{9}{x \cdot 5} + \frac{C}{x^{6}}$

Step 6.8.3.1.4

Move $5$ to the left of $x$ .

$v = - \frac{9}{5 x} + \frac{C}{x^{6}}$

Step 7

Substitute $y^{- 3}$ for $v$ .

$y^{- 3} = - \frac{9}{5 x} + \frac{C}{x^{6}}$