Calculus Examples

$6 \frac{d y}{d x} - 2 y = x y^{4}$

Step 1

Rewrite the differential equation to fit the Bernoulli technique.

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Step 1.1

Divide each term in $6 \frac{d y}{d x} - 2 y = x y^{4}$ by $6$ .

$\frac{6 \frac{d y}{d x}}{6} + \frac{- 2 y}{6} = \frac{x y^{4}}{6}$

Step 1.2

Cancel the common factor of $6$ .

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Step 1.2.1

Cancel the common factor.

$\frac{6 \frac{d y}{d x}}{6} + \frac{- 2 y}{6} = \frac{x y^{4}}{6}$

Step 1.2.2

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

$\frac{d y}{d x} + \frac{- 2 y}{6} = \frac{x y^{4}}{6}$

Step 1.3

Cancel the common factor of $- 2$ and $6$ .

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Step 1.3.1

Factor $2$ out of $- 2 y$ .

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

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $2$ out of $6$ .

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

Step 1.3.2.2

Cancel the common factor.

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

Step 1.3.2.3

Rewrite the expression.

$\frac{d y}{d x} + \frac{- y}{3} = \frac{x y^{4}}{6}$

Step 1.4

Reorder terms.

$\frac{d y}{d x} + \frac{- 1}{3} y = \frac{1}{6} x y^{4}$

Step 2

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

$v = y^{- 3}$

Step 3

Solve the equation for $y$ .

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

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

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

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

Step 5.4

Differentiate using the Constant Rule.

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Step 5.4.1

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 5.4.2

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

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Step 5.4.2.1

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

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

Step 5.4.2.2

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 5.4.3

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

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

Step 5.4.4

Simplify the expression.

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Step 5.4.4.1

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

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

Step 5.4.4.2

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 5.4.4.3

Move the negative in front of the fraction.

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

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

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

Step 5.5.3

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 5.6

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 5.7

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 5.8

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 5.9

Simplify the numerator.

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Step 5.9.1

Multiply $- 1$ by $3$ .

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

Step 5.9.2

Subtract $3$ from $1$ .

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

Step 5.10

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 5.11

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 5.12

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 5.13

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

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

Step 5.14

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

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

Step 5.15

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 5.16

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 5.17

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

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Step 5.17.1

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

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

Step 5.17.2

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 5.17.3

Combine the numerators over the common denominator.

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

Step 5.17.4

Add $2$ and $2$ .

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

Step 6

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 $\frac{d y}{d x} + \frac{- 1}{3} y = \frac{1}{6} x y^{4}$ .

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

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 $- \frac{\frac{d v}{d x}}{3 v^{\frac{4}{3}}} + \frac{- 1}{3} v^{- \frac{1}{3}} = \frac{1}{6} x {(v^{- \frac{1}{3}})}^{4}$ by $- (3 v^{\frac{4}{3}})$ to eliminate the fractions.

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Step 7.1.1.1

Multiply each term in $- \frac{\frac{d v}{d x}}{3 v^{\frac{4}{3}}} + \frac{- 1}{3} v^{- \frac{1}{3}} = \frac{1}{6} x {(v^{- \frac{1}{3}})}^{4}$ by $- (3 v^{\frac{4}{3}})$ .

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

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

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Step 7.1.1.2.1.1.1

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

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

Step 7.1.1.2.1.1.2

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

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

Step 7.1.1.2.1.1.3

Cancel the common factor.

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

Step 7.1.1.2.1.1.4

Rewrite the expression.

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

Step 7.1.1.2.1.2

Multiply $- 1$ by $- 1$ .

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

Step 7.1.1.2.1.3

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

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

Step 7.1.1.2.1.4

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

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Step 7.1.1.2.1.4.1

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

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

Step 7.1.1.2.1.4.2

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

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

Step 7.1.1.2.1.4.3

Combine the numerators over the common denominator.

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

Step 7.1.1.2.1.4.4

Subtract $1$ from $4$ .

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

Step 7.1.1.2.1.4.5

Divide $3$ by $3$ .

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

Step 7.1.1.2.1.5

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

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

Step 7.1.1.2.1.6

Move the negative in front of the fraction.

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

Step 7.1.1.2.1.7

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

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

Step 7.1.1.2.1.8

Cancel the common factor of $3$ .

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Step 7.1.1.2.1.8.1

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

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

Step 7.1.1.2.1.8.2

Factor $3$ out of $- 1 \cdot (3)$ .

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

Step 7.1.1.2.1.8.3

Cancel the common factor.

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

Step 7.1.1.2.1.8.4

Rewrite the expression.

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

Step 7.1.1.2.1.9

Multiply $- 1$ by $- 1$ .

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

Step 7.1.1.2.1.10

Multiply $v$ by $1$ .

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

Step 7.1.1.3

Simplify the right side.

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Step 7.1.1.3.1

Combine $\frac{1}{6}$ and $x$ .

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

Step 7.1.1.3.2

Rewrite using the commutative property of multiplication.

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

Step 7.1.1.3.3

Cancel the common factor of $3$ .

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Step 7.1.1.3.3.1

Factor $3$ out of $- 1 \cdot 3$ .

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

Step 7.1.1.3.3.2

Factor $3$ out of $6$ .

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

Step 7.1.1.3.3.3

Cancel the common factor.

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

Step 7.1.1.3.3.4

Rewrite the expression.

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

Step 7.1.1.3.4

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

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Step 7.1.1.3.4.1

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

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

Step 7.1.1.3.4.2

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

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Step 7.1.1.3.4.2.1

Multiply $4$ by $- 1$ .

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

Step 7.1.1.3.4.2.2

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

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

Step 7.1.1.3.4.3

Move the negative in front of the fraction.

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

Step 7.1.1.3.5

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

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Step 7.1.1.3.5.1

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

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

Step 7.1.1.3.5.2

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

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

Step 7.1.1.3.5.3

Combine the numerators over the common denominator.

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

Step 7.1.1.3.5.4

Subtract $4$ from $4$ .

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

Step 7.1.1.3.5.5

Divide $0$ by $3$ .

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

Step 7.1.1.3.6

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

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

Step 7.1.2

Rewrite the equation with isolated coefficients.

$\frac{d v}{d x} + v = - \frac{1}{2} x$

Step 7.2

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

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Step 7.2.1

Set up the integration.

$e^{\int d x}$

Step 7.2.2

Apply the constant rule.

$e^{x + C}$

Step 7.2.3

Remove the constant of integration.

$e^{x}$

Step 7.3

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

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Step 7.3.1

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

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

Step 7.3.2

Rewrite using the commutative property of multiplication.

$e^{x} \frac{d v}{d x} + e^{x} v = - \frac{1}{2} e^{x} x$

Step 7.3.3

Combine $e^{x}$ and $\frac{1}{2}$ .

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

Step 7.3.4

Combine $x$ and $\frac{e^{x}}{2}$ .

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

Step 7.3.5

Reorder factors in $e^{x} \frac{d v}{d x} + e^{x} v = - \frac{x e^{x}}{2}$ .

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

Step 7.4

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

$\frac{d}{d x} [e^{x} v] = - \frac{x e^{x}}{2}$

Step 7.5

Set up an integral on each side.

$\int \frac{d}{d x} [e^{x} v] d x = \int - \frac{x e^{x}}{2} d x$

Step 7.6

Integrate the left side.

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

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.

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

Step 7.7.2

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 7.7.3

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{x}$ .

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

Step 7.7.4

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$e^{x} v = - \frac{1}{2} (x e^{x} - (e^{x} + C))$

Step 7.7.5

Simplify.

$e^{x} v = - \frac{1}{2} (x e^{x} - e^{x}) + C$

Step 7.8

Divide each term in $e^{x} v = - \frac{1}{2} (x e^{x} - e^{x}) + C$ by $e^{x}$ and simplify.

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Step 7.8.1

Divide each term in $e^{x} v = - \frac{1}{2} (x e^{x} - e^{x}) + C$ by $e^{x}$ .

$\frac{e^{x} v}{e^{x}} = \frac{- \frac{1}{2} (x e^{x} - e^{x})}{e^{x}} + \frac{C}{e^{x}}$

Step 7.8.2

Simplify the left side.

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Step 7.8.2.1

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

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Step 7.8.2.1.1

Cancel the common factor.

$\frac{e^{x} v}{e^{x}} = \frac{- \frac{1}{2} (x e^{x} - e^{x})}{e^{x}} + \frac{C}{e^{x}}$

Step 7.8.2.1.2

Divide $v$ by $1$ .

$v = \frac{- \frac{1}{2} (x e^{x} - e^{x})}{e^{x}} + \frac{C}{e^{x}}$

Step 7.8.3

Simplify the right side.

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Step 7.8.3.1

Combine into one fraction.

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Step 7.8.3.1.1

Combine the numerators over the common denominator.

$v = \frac{- \frac{1}{2} (x e^{x} - e^{x}) + C}{e^{x}}$

Step 7.8.3.1.2

Reorder factors in $- \frac{1}{2} (x e^{x} - e^{x}) + C$ .

$v = \frac{- (x e^{x} - e^{x}) \frac{1}{2} + C}{e^{x}}$

Step 7.8.3.2

Simplify the numerator.

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Step 7.8.3.2.1

Apply the distributive property.

$v = \frac{(- (x e^{x}) - - e^{x}) \frac{1}{2} + C}{e^{x}}$

Step 7.8.3.2.2

Multiply $- - e^{x}$ .

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Step 7.8.3.2.2.1

Multiply $- 1$ by $- 1$ .

$v = \frac{(- x e^{x} + 1 e^{x}) \frac{1}{2} + C}{e^{x}}$

Step 7.8.3.2.2.2

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

$v = \frac{(- x e^{x} + e^{x}) \frac{1}{2} + C}{e^{x}}$

Step 7.8.3.2.3

Apply the distributive property.

$v = \frac{- x e^{x} \frac{1}{2} + e^{x} \frac{1}{2} + C}{e^{x}}$

Step 7.8.3.2.4

Multiply $- x e^{x} \frac{1}{2}$ .

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Step 7.8.3.2.4.1

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

$v = \frac{- \frac{x}{2} e^{x} + e^{x} \frac{1}{2} + C}{e^{x}}$

Step 7.8.3.2.4.2

Combine $e^{x}$ and $\frac{x}{2}$ .

$v = \frac{- \frac{e^{x} x}{2} + e^{x} \frac{1}{2} + C}{e^{x}}$

Step 7.8.3.2.5

Combine $e^{x}$ and $\frac{1}{2}$ .

$v = \frac{- \frac{e^{x} x}{2} + \frac{e^{x}}{2} + C}{e^{x}}$

Step 7.8.3.2.6

Combine the numerators over the common denominator.

$v = \frac{\frac{- e^{x} x + e^{x}}{2} + C}{e^{x}}$

Step 7.8.3.2.7

Factor $e^{x}$ out of $- e^{x} x + e^{x}$ .

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Step 7.8.3.2.7.1

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

$v = \frac{\frac{e^{x} (- x) + e^{x}}{2} + C}{e^{x}}$

Step 7.8.3.2.7.2

Multiply by $1$ .

$v = \frac{\frac{e^{x} (- x) + e^{x} \cdot 1}{2} + C}{e^{x}}$

Step 7.8.3.2.7.3

Factor $e^{x}$ out of $e^{x} (- x) + e^{x} \cdot 1$ .

$v = \frac{\frac{e^{x} (- x + 1)}{2} + C}{e^{x}}$

Step 7.8.3.2.8

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

$v = \frac{\frac{e^{x} (- x + 1)}{2} + C \cdot \frac{2}{2}}{e^{x}}$

Step 7.8.3.2.9

Combine $C$ and $\frac{2}{2}$ .

$v = \frac{\frac{e^{x} (- x + 1)}{2} + \frac{C \cdot 2}{2}}{e^{x}}$

Step 7.8.3.2.10

Combine the numerators over the common denominator.

$v = \frac{\frac{e^{x} (- x + 1) + C \cdot 2}{2}}{e^{x}}$

Step 7.8.3.2.11

Simplify the numerator.

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Step 7.8.3.2.11.1

Apply the distributive property.

$v = \frac{\frac{e^{x} (- x) + e^{x} \cdot 1 + C \cdot 2}{2}}{e^{x}}$

Step 7.8.3.2.11.2

Rewrite using the commutative property of multiplication.

$v = \frac{\frac{- e^{x} x + e^{x} \cdot 1 + C \cdot 2}{2}}{e^{x}}$

Step 7.8.3.2.11.3

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

$v = \frac{\frac{- e^{x} x + e^{x} + C \cdot 2}{2}}{e^{x}}$

Step 7.8.3.2.11.4

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

$v = \frac{\frac{- e^{x} x + e^{x} + 2 C}{2}}{e^{x}}$

Step 7.8.3.3

Multiply the numerator by the reciprocal of the denominator.

$v = \frac{- e^{x} x + e^{x} + 2 C}{2} \cdot \frac{1}{e^{x}}$

Step 7.8.3.4

Multiply $\frac{- e^{x} x + e^{x} + 2 C}{2}$ by $\frac{1}{e^{x}}$ .

$v = \frac{- e^{x} x + e^{x} + 2 C}{2 e^{x}}$

Step 7.8.3.5

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

$v = \frac{- (e^{x} x) + e^{x} + 2 C}{2 e^{x}}$

Step 7.8.3.6

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

$v = \frac{- (e^{x} x) - 1 (- e^{x}) + 2 C}{2 e^{x}}$

Step 7.8.3.7

Factor $- 1$ out of $- (e^{x} x) - 1 (- e^{x})$ .

$v = \frac{- (e^{x} x - e^{x}) + 2 C}{2 e^{x}}$

Step 7.8.3.8

Factor $- 1$ out of $2 C$ .

$v = \frac{- (e^{x} x - e^{x}) - (- 2 C)}{2 e^{x}}$

Step 7.8.3.9

Factor $- 1$ out of $- (e^{x} x - e^{x}) - (- 2 C)$ .

$v = \frac{- (e^{x} x - e^{x} - 2 C)}{2 e^{x}}$

Step 7.8.3.10

Simplify the expression.

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Step 7.8.3.10.1

Rewrite $- (e^{x} x - e^{x} - 2 C)$ as $- 1 (e^{x} x - e^{x} - 2 C)$ .

$v = \frac{- 1 (e^{x} x - e^{x} - 2 C)}{2 e^{x}}$

Step 7.8.3.10.2

Move the negative in front of the fraction.

$v = - \frac{e^{x} x - e^{x} - 2 C}{2 e^{x}}$

Step 7.8.3.10.3

Reorder factors in $- \frac{e^{x} x - e^{x} - 2 C}{2 e^{x}}$ .

$v = - \frac{x e^{x} - e^{x} - 2 C}{2 e^{x}}$

Step 8

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

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