Calculus Examples

$2 \frac{d y}{d x} - \frac{y}{x} = 5 x^{3} y^{3}$

Step 1

Rewrite the differential equation to fit the Bernoulli technique.

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Step 1.1

Divide each term in $2 \frac{d y}{d x} - \frac{y}{x} = 5 x^{3} y^{3}$ by $2$ .

$\frac{2 \frac{d y}{d x}}{2} + \frac{- \frac{y}{x}}{2} = \frac{5 x^{3} y^{3}}{2}$

Step 1.2

Cancel the common factor of $2$ .

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Step 1.2.1

Cancel the common factor.

$\frac{2 \frac{d y}{d x}}{2} + \frac{- \frac{y}{x}}{2} = \frac{5 x^{3} y^{3}}{2}$

Step 1.2.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} + \frac{- \frac{y}{x}}{2} = \frac{5 x^{3} y^{3}}{2}$

Step 1.3

Reorder terms.

$\frac{d y}{d x} + \frac{- 1}{2} \cdot \frac{y}{x} = \frac{5}{2} x^{3} y^{3}$

Step 1.4

Factor $y$ out of $\frac{- 1}{2} \cdot \frac{y}{x}$ .

$\frac{d y}{d x} + y (\frac{- 1}{2} \cdot \frac{1}{x}) = \frac{5}{2} x^{3} y^{3}$

Step 1.5

Reorder $y$ and $\frac{- 1}{2} \cdot \frac{1}{x}$ .

$\frac{d y}{d x} + \frac{- 1}{2} \cdot \frac{1}{x} y = \frac{5}{2} x^{3} y^{3}$

Step 2

To solve the differential equation, let $v = y^{1 - n}$ where $n$ is the exponent of $y^{3}$ .

$v = y^{- 2}$

Step 3

Solve the equation for $y$ .

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

Step 4

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

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

Step 5

Take the derivative of $v^{- \frac{1}{2}}$ with respect to $x$ .

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Step 5.1

Take the derivative of $v^{- \frac{1}{2}}$ .

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

Step 5.2

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

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

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 $g (x) = v^{\frac{1}{2}}$ .

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

Step 5.4

Simplify the expression.

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Step 5.4.1

Multiply $- 1$ by $1$ .

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

Step 5.4.2

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

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Step 5.4.2.1

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

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

Step 5.4.2.2

Cancel the common factor of $2$ .

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Step 5.4.2.2.1

Cancel the common factor.

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

Step 5.4.2.2.2

Rewrite the expression.

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

Step 5.5

Simplify.

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

Step 5.6

Differentiate using the Constant Rule.

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Step 5.6.1

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

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

Step 5.6.2

Simplify the expression.

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Step 5.6.2.1

Multiply $v^{\frac{1}{2}}$ by $0$ .

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

Step 5.6.2.2

Subtract $\frac{d}{d x} [v^{\frac{1}{2}}]$ from $0$ .

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

Step 5.6.2.3

Move the negative in front of the fraction.

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

Step 5.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = v$ .

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Step 5.7.1

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

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

Step 5.7.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{2}$ .

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

Step 5.7.3

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

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

Step 5.8

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

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

Step 5.9

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

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

Step 5.10

Combine the numerators over the common denominator.

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

Step 5.11

Simplify the numerator.

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Step 5.11.1

Multiply $- 1$ by $2$ .

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

Step 5.11.2

Subtract $2$ from $1$ .

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

Step 5.12

Move the negative in front of the fraction.

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

Step 5.13

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

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

Step 5.14

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

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

Step 5.15

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

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

Step 5.16

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

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

Step 5.17

Rewrite $\frac{\frac{v'}{2 v^{\frac{1}{2}}}}{v}$ as a product.

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

Step 5.18

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

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

Step 5.19

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

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

Step 5.20

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

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

Step 5.21

Write $1$ as a fraction with a common denominator.

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

Step 5.22

Combine the numerators over the common denominator.

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

Step 5.23

Add $2$ and $1$ .

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

Step 6

Substitute $- \frac{v'}{2 v^{\frac{3}{2}}}$ for $\frac{d y}{d x}$ and $v^{- \frac{1}{2}}$ for $y$ in the original equation $\frac{d y}{d x} + \frac{- 1}{2} \cdot \frac{1}{x} y = \frac{5}{2} x^{3} y^{3}$ .

$- \frac{v'}{2 v^{\frac{3}{2}}} + \frac{- 1}{2} \cdot \frac{1}{x} v^{- \frac{1}{2}} = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3}$

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 $- \frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}} + \frac{- 1}{2} \cdot \frac{1}{x} v^{- \frac{1}{2}} = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3}$ by $- (2 v^{\frac{3}{2}})$ to eliminate the fractions.

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Step 7.1.1.1

Multiply each term in $- \frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}} + \frac{- 1}{2} \cdot \frac{1}{x} v^{- \frac{1}{2}} = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3}$ by $- (2 v^{\frac{3}{2}})$ .

$- \frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}} (- (2 v^{\frac{3}{2}})) + \frac{- 1}{2} \cdot \frac{1}{x} v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

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

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Step 7.1.1.2.1.1.1

Move the leading negative in $- \frac{\frac{d v}{d x}}{2 v^{\frac{3}{2}}}$ into the numerator.

$\frac{- \frac{d v}{d x}}{2 v^{\frac{3}{2}}} (- (2 v^{\frac{3}{2}})) + \frac{- 1}{2} \cdot \frac{1}{x} v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.1.2

Factor $2 v^{\frac{3}{2}}$ out of $- (2 v^{\frac{3}{2}})$ .

$\frac{- \frac{d v}{d x}}{2 v^{\frac{3}{2}}} (2 v^{\frac{3}{2}} (- 1)) + \frac{- 1}{2} \cdot \frac{1}{x} v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.1.3

Cancel the common factor.

$\frac{- \frac{d v}{d x}}{2 v^{\frac{3}{2}}} (2 v^{\frac{3}{2}} \cdot - 1) + \frac{- 1}{2} \cdot \frac{1}{x} v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.1.4

Rewrite the expression.

$- \frac{d v}{d x} \cdot - 1 + \frac{- 1}{2} \cdot \frac{1}{x} v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.2

Multiply $- 1$ by $- 1$ .

$1 \frac{d v}{d x} + \frac{- 1}{2} \cdot \frac{1}{x} v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.3

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

$\frac{d v}{d x} + \frac{- 1}{2} \cdot \frac{1}{x} v^{- \frac{1}{2}} (- (2 v^{\frac{3}{2}})) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.4

Multiply $v^{- \frac{1}{2}}$ by $v^{\frac{3}{2}}$ by adding the exponents.

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Step 7.1.1.2.1.4.1

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

$\frac{d v}{d x} + \frac{- 1}{2} \cdot \frac{1}{x} (v^{\frac{3}{2}} v^{- \frac{1}{2}}) (- 1 \cdot (2)) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.4.2

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

$\frac{d v}{d x} + \frac{- 1}{2} \cdot \frac{1}{x} v^{\frac{3}{2} - \frac{1}{2}} (- 1 \cdot (2)) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.4.3

Combine the numerators over the common denominator.

$\frac{d v}{d x} + \frac{- 1}{2} \cdot \frac{1}{x} v^{\frac{3 - 1}{2}} (- 1 \cdot (2)) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.4.4

Subtract $1$ from $3$ .

$\frac{d v}{d x} + \frac{- 1}{2} \cdot \frac{1}{x} v^{\frac{2}{2}} (- 1 \cdot (2)) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.4.5

Divide $2$ by $2$ .

$\frac{d v}{d x} + \frac{- 1}{2} \cdot \frac{1}{x} v^{1} (- 1 \cdot (2)) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.5

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

$\frac{d v}{d x} + \frac{- 1}{2} \cdot \frac{1}{x} v (- 1 \cdot (2)) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.6

Move the negative in front of the fraction.

$\frac{d v}{d x} - \frac{1}{2} \cdot \frac{1}{x} v (- 1 \cdot (2)) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.7

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

$\frac{d v}{d x} - \frac{1}{x \cdot 2} v (- 1 \cdot (2)) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.8

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

$\frac{d v}{d x} - \frac{1}{2 \cdot x} v (- 1 \cdot (2)) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.9

Combine $v$ and $\frac{1}{2 x}$ .

$\frac{d v}{d x} - \frac{v}{2 x} (- 1 \cdot (2)) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.10

Cancel the common factor of $2$ .

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Step 7.1.1.2.1.10.1

Move the leading negative in $- \frac{v}{2 x}$ into the numerator.

$\frac{d v}{d x} + \frac{- v}{2 x} (- 1 \cdot (2)) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.10.2

Factor $2$ out of $2 x$ .

$\frac{d v}{d x} + \frac{- v}{2 (x)} (- 1 \cdot (2)) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.10.3

Factor $2$ out of $- 1 \cdot (2)$ .

$\frac{d v}{d x} + \frac{- v}{2 (x)} (2 \cdot - 1) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.10.4

Cancel the common factor.

$\frac{d v}{d x} + \frac{- v}{2 x} (2 \cdot - 1) = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.10.5

Rewrite the expression.

$\frac{d v}{d x} + \frac{- v}{x} \cdot - 1 = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.11

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

$\frac{d v}{d x} + \frac{- v \cdot - 1}{x} = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.12

Multiply $- 1$ by $- 1$ .

$\frac{d v}{d x} + \frac{1 v}{x} = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.2.1.13

Multiply $v$ by $1$ .

$\frac{d v}{d x} + \frac{v}{x} = \frac{5}{2} x^{3} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.3

Simplify the right side.

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Step 7.1.1.3.1

Combine $\frac{5}{2}$ and $x^{3}$ .

$\frac{d v}{d x} + \frac{v}{x} = \frac{5 x^{3}}{2} {(v^{- \frac{1}{2}})}^{3} (- (2 v^{\frac{3}{2}}))$

Step 7.1.1.3.2

Rewrite using the commutative property of multiplication.

$\frac{d v}{d x} + \frac{v}{x} = - 1 \cdot 2 \frac{5 x^{3}}{2} {(v^{- \frac{1}{2}})}^{3} v^{\frac{3}{2}}$

Step 7.1.1.3.3

Cancel the common factor of $2$ .

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Step 7.1.1.3.3.1

Factor $2$ out of $- 1 \cdot 2$ .

$\frac{d v}{d x} + \frac{v}{x} = 2 \cdot - 1 \frac{5 x^{3}}{2} {(v^{- \frac{1}{2}})}^{3} v^{\frac{3}{2}}$

Step 7.1.1.3.3.2

Cancel the common factor.

$\frac{d v}{d x} + \frac{v}{x} = 2 \cdot - 1 \frac{5 x^{3}}{2} {(v^{- \frac{1}{2}})}^{3} v^{\frac{3}{2}}$

Step 7.1.1.3.3.3

Rewrite the expression.

$\frac{d v}{d x} + \frac{v}{x} = - (5 x^{3}) {(v^{- \frac{1}{2}})}^{3} v^{\frac{3}{2}}$

Step 7.1.1.3.4

Simplify the expression.

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Step 7.1.1.3.4.1

Multiply $5$ by $- 1$ .

$\frac{d v}{d x} + \frac{v}{x} = - 5 x^{3} {(v^{- \frac{1}{2}})}^{3} v^{\frac{3}{2}}$

Step 7.1.1.3.4.2

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

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Step 7.1.1.3.4.2.1

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

$\frac{d v}{d x} + \frac{v}{x} = - 5 x^{3} v^{- \frac{1}{2} \cdot 3} v^{\frac{3}{2}}$

Step 7.1.1.3.4.2.2

Multiply $- \frac{1}{2} \cdot 3$ .

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Step 7.1.1.3.4.2.2.1

Multiply $3$ by $- 1$ .

$\frac{d v}{d x} + \frac{v}{x} = - 5 x^{3} v^{- 3 (\frac{1}{2})} v^{\frac{3}{2}}$

Step 7.1.1.3.4.2.2.2

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

$\frac{d v}{d x} + \frac{v}{x} = - 5 x^{3} v^{\frac{- 3}{2}} v^{\frac{3}{2}}$

Step 7.1.1.3.4.2.3

Move the negative in front of the fraction.

$\frac{d v}{d x} + \frac{v}{x} = - 5 x^{3} v^{- \frac{3}{2}} v^{\frac{3}{2}}$

Step 7.1.1.3.5

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

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Step 7.1.1.3.5.1

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

$\frac{d v}{d x} + \frac{v}{x} = - 5 x^{3} (v^{\frac{3}{2}} v^{- \frac{3}{2}})$

Step 7.1.1.3.5.2

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

$\frac{d v}{d x} + \frac{v}{x} = - 5 x^{3} v^{\frac{3}{2} - \frac{3}{2}}$

Step 7.1.1.3.5.3

Combine the numerators over the common denominator.

$\frac{d v}{d x} + \frac{v}{x} = - 5 x^{3} v^{\frac{3 - 3}{2}}$

Step 7.1.1.3.5.4

Subtract $3$ from $3$ .

$\frac{d v}{d x} + \frac{v}{x} = - 5 x^{3} v^{\frac{0}{2}}$

Step 7.1.1.3.5.5

Divide $0$ by $2$ .

$\frac{d v}{d x} + \frac{v}{x} = - 5 x^{3} v^{0}$

Step 7.1.1.3.6

Simplify $- 5 x^{3} v^{0}$ .

$\frac{d v}{d x} + \frac{v}{x} = - 5 x^{3}$

Step 7.1.2

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

$\frac{d v}{d x} + v \frac{1}{x} = - 5 x^{3}$

Step 7.1.3

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

$\frac{d v}{d x} + \frac{1}{x} v = - 5 x^{3}$

Step 7.2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{1}{x}$ .

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Step 7.2.1

Set up the integration.

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

Step 7.2.2

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

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

Step 7.2.3

Remove the constant of integration.

$e^{\ln (x)}$

Step 7.2.4

Exponentiation and log are inverse functions.

$x$

Step 7.3

Multiply each term by the integrating factor $x$ .

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Step 7.3.1

Multiply each term by $x$ .

$x \frac{d v}{d x} + x (\frac{1}{x} v) = x (- 5 x^{3})$

Step 7.3.2

Simplify each term.

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Step 7.3.2.1

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

$x \frac{d v}{d x} + x \frac{v}{x} = x (- 5 x^{3})$

Step 7.3.2.2

Cancel the common factor of $x$ .

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Step 7.3.2.2.1

Cancel the common factor.

$x \frac{d v}{d x} + x \frac{v}{x} = x (- 5 x^{3})$

Step 7.3.2.2.2

Rewrite the expression.

$x \frac{d v}{d x} + v = x (- 5 x^{3})$

Step 7.3.3

Rewrite using the commutative property of multiplication.

$x \frac{d v}{d x} + v = - 5 x \cdot x^{3}$

Step 7.3.4

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 7.3.4.1

Move $x^{3}$ .

$x \frac{d v}{d x} + v = - 5 (x^{3} x)$

Step 7.3.4.2

Multiply $x^{3}$ by $x$ .

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Step 7.3.4.2.1

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

$x \frac{d v}{d x} + v = - 5 (x^{3} x^{1})$

Step 7.3.4.2.2

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

$x \frac{d v}{d x} + v = - 5 x^{3 + 1}$

Step 7.3.4.3

Add $3$ and $1$ .

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

Step 7.4

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

$\frac{d}{d x} [x v] = - 5 x^{4}$

Step 7.5

Set up an integral on each side.

$\int \frac{d}{d x} [x v] d x = \int - 5 x^{4} d x$

Step 7.6

Integrate the left side.

$x v = \int - 5 x^{4} d x$

Step 7.7

Integrate the right side.

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Step 7.7.1

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

$x v = - 5 \int x^{4} d x$

Step 7.7.2

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

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

Step 7.7.3

Simplify the answer.

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Step 7.7.3.1

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

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

Step 7.7.3.2

Simplify.

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Step 7.7.3.2.1

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

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

Step 7.7.3.2.2

Cancel the common factor of $- 5$ and $5$ .

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Step 7.7.3.2.2.1

Factor $5$ out of $- 5$ .

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

Step 7.7.3.2.2.2

Cancel the common factors.

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Step 7.7.3.2.2.2.1

Factor $5$ out of $5$ .

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

Step 7.7.3.2.2.2.2

Cancel the common factor.

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

Step 7.7.3.2.2.2.3

Rewrite the expression.

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

Step 7.7.3.2.2.2.4

Divide $- 1$ by $1$ .

$x v = - x^{5} + C$

Step 7.8

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

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Step 7.8.1

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

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

Step 7.8.2

Simplify the left side.

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Step 7.8.2.1

Cancel the common factor of $x$ .

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Step 7.8.2.1.1

Cancel the common factor.

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

Step 7.8.2.1.2

Divide $v$ by $1$ .

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

Step 7.8.3

Simplify the right side.

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Step 7.8.3.1

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

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Step 7.8.3.1.1

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

$v = \frac{x (- x^{4})}{x} + \frac{C}{x}$

Step 7.8.3.1.2

Cancel the common factors.

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Step 7.8.3.1.2.1

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

$v = \frac{x (- x^{4})}{x^{1}} + \frac{C}{x}$

Step 7.8.3.1.2.2

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

$v = \frac{x (- x^{4})}{x \cdot 1} + \frac{C}{x}$

Step 7.8.3.1.2.3

Cancel the common factor.

$v = \frac{x (- x^{4})}{x \cdot 1} + \frac{C}{x}$

Step 7.8.3.1.2.4

Rewrite the expression.

$v = \frac{- x^{4}}{1} + \frac{C}{x}$

Step 7.8.3.1.2.5

Divide $- x^{4}$ by $1$ .

$v = - x^{4} + \frac{C}{x}$

Step 8

Substitute $y^{- 2}$ for $v$ .

$y^{- 2} = - x^{4} + \frac{C}{x}$