Calculus Examples

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

Step 1

Rewrite the differential equation to fit the Bernoulli technique.

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Step 1.1

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

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

Step 1.2

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

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

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

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

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

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Step 7.1.1.1

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

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

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.

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

Step 7.1.1.2.1.1.2

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

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

Step 7.1.1.2.1.1.3

Cancel the common factor.

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

Step 7.1.1.2.1.1.4

Rewrite the expression.

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

Step 7.1.1.2.1.2

Multiply $- 1$ by $- 1$ .

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

Step 7.1.1.2.1.3

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

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

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

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

Step 7.1.1.2.1.4.2

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

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

Step 7.1.1.2.1.4.3

Combine the numerators over the common denominator.

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

Step 7.1.1.2.1.4.4

Subtract $1$ from $3$ .

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

Step 7.1.1.2.1.4.5

Divide $2$ by $2$ .

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

Step 7.1.1.2.1.5

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

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

Step 7.1.1.2.1.6

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

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

Step 7.1.1.2.1.7

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

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

Step 7.1.1.2.1.8

Multiply $- 1$ by $2$ .

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

Step 7.1.1.2.1.9

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

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Step 7.1.1.2.1.9.1

Multiply $- 2$ by $- 1$ .

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

Step 7.1.1.2.1.9.2

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

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

Step 7.1.1.2.1.9.3

Multiply $2$ by $2$ .

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

Step 7.1.1.3

Simplify the right side.

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Step 7.1.1.3.1

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

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Step 7.1.1.3.1.1

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

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

Step 7.1.1.3.1.2

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

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Step 7.1.1.3.1.2.1

Multiply $3$ by $- 1$ .

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

Step 7.1.1.3.1.2.2

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

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

Step 7.1.1.3.1.3

Move the negative in front of the fraction.

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

Step 7.1.1.3.2

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

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Step 7.1.1.3.2.1

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

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

Step 7.1.1.3.2.2

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

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

Step 7.1.1.3.2.3

Combine the numerators over the common denominator.

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

Step 7.1.1.3.2.4

Subtract $3$ from $3$ .

$\frac{d v}{d x} + \frac{4 v}{x} = 2 v^{\frac{0}{2}} (- 1 \cdot (2))$

Step 7.1.1.3.2.5

Divide $0$ by $2$ .

$\frac{d v}{d x} + \frac{4 v}{x} = 2 v^{0} (- 1 \cdot (2))$

Step 7.1.1.3.3

Simplify $2 v^{0} (- 1 \cdot (2))$ .

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

Step 7.1.1.3.4

Multiply $2 (- 1 \cdot (2))$ .

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Step 7.1.1.3.4.1

Multiply $- 1$ by $2$ .

$\frac{d v}{d x} + \frac{4 v}{x} = 2 \cdot - 2$

Step 7.1.1.3.4.2

Multiply $2$ by $- 2$ .

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

Step 7.1.2

Factor $v$ out of $\frac{4 v}{x}$ .

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

Step 7.1.3

Reorder $v$ and $\frac{4}{x}$ .

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

Step 7.2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{4}{x}$ .

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Step 7.2.1

Set up the integration.

$e^{\int \frac{4}{x} d x}$

Step 7.2.2

Integrate $\frac{4}{x}$ .

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Step 7.2.2.1

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

$e^{4 \int \frac{1}{x} d x}$

Step 7.2.2.2

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

$e^{4 (\ln (| x |) + C)}$

Step 7.2.2.3

Simplify.

$e^{4 \ln (| x |) + C}$

Step 7.2.3

Remove the constant of integration.

$e^{4 \ln (x)}$

Step 7.2.4

Use the logarithmic power rule.

$e^{\ln (x^{4})}$

Step 7.2.5

Exponentiation and log are inverse functions.

$x^{4}$

Step 7.3

Multiply each term by the integrating factor $x^{4}$ .

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Step 7.3.1

Multiply each term by $x^{4}$ .

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

Step 7.3.2

Simplify each term.

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Step 7.3.2.1

Combine $\frac{4}{x}$ and $v$ .

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

Step 7.3.2.2

Cancel the common factor of $x$ .

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Step 7.3.2.2.1

Factor $x$ out of $x^{4}$ .

$x^{4} \frac{d v}{d x} + x \cdot x^{3} \frac{4 v}{x} = x^{4} \cdot - 4$

Step 7.3.2.2.2

Cancel the common factor.

$x^{4} \frac{d v}{d x} + x \cdot x^{3} \frac{4 v}{x} = x^{4} \cdot - 4$

Step 7.3.2.2.3

Rewrite the expression.

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

Step 7.3.2.3

Rewrite using the commutative property of multiplication.

$x^{4} \frac{d v}{d x} + 4 x^{3} v = x^{4} \cdot - 4$

Step 7.3.3

Move $- 4$ to the left of $x^{4}$ .

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

Step 7.4

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

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

Step 7.5

Set up an integral on each side.

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

Step 7.6

Integrate the left side.

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

Step 7.7

Integrate the right side.

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Step 7.7.1

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

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

Step 7.7.2

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

$x^{4} v = - 4 (\frac{1}{5} x^{5} + C)$

Step 7.7.3

Simplify the answer.

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Step 7.7.3.1

Rewrite $- 4 (\frac{1}{5} x^{5} + C)$ as $- 4 (\frac{1}{5}) x^{5} + C$ .

$x^{4} v = - 4 (\frac{1}{5}) x^{5} + C$

Step 7.7.3.2

Simplify.

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Step 7.7.3.2.1

Combine $- 4$ and $\frac{1}{5}$ .

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

Step 7.7.3.2.2

Move the negative in front of the fraction.

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

Step 7.8

Divide each term in $x^{4} v = - \frac{4}{5} x^{5} + C$ by $x^{4}$ and simplify.

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Step 7.8.1

Divide each term in $x^{4} v = - \frac{4}{5} x^{5} + C$ by $x^{4}$ .

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

Step 7.8.2

Simplify the left side.

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Step 7.8.2.1

Cancel the common factor of $x^{4}$ .

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Step 7.8.2.1.1

Cancel the common factor.

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

Step 7.8.2.1.2

Divide $v$ by $1$ .

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

Step 7.8.3

Simplify the right side.

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Step 7.8.3.1

Simplify each term.

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Step 7.8.3.1.1

Cancel the common factor of $x^{5}$ and $x^{4}$ .

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Step 7.8.3.1.1.1

Factor $x^{4}$ out of $- \frac{4}{5} x^{5}$ .

$v = \frac{x^{4} (- \frac{4}{5} x)}{x^{4}} + \frac{C}{x^{4}}$

Step 7.8.3.1.1.2

Cancel the common factors.

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Step 7.8.3.1.1.2.1

Multiply by $1$ .

$v = \frac{x^{4} (- \frac{4}{5} x)}{x^{4} \cdot 1} + \frac{C}{x^{4}}$

Step 7.8.3.1.1.2.2

Cancel the common factor.

$v = \frac{x^{4} (- \frac{4}{5} x)}{x^{4} \cdot 1} + \frac{C}{x^{4}}$

Step 7.8.3.1.1.2.3

Rewrite the expression.

$v = \frac{- \frac{4}{5} x}{1} + \frac{C}{x^{4}}$

Step 7.8.3.1.1.2.4

Divide $- \frac{4}{5} x$ by $1$ .

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

Step 7.8.3.1.2

Combine $x$ and $\frac{4}{5}$ .

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

Step 7.8.3.1.3

Move $4$ to the left of $x$ .

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

Step 8

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

$y^{- 2} = - \frac{4 x}{5} + \frac{C}{x^{4}}$