Calculus Examples

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

Step 1

Rewrite the differential equation as a function of $\frac{x}{y}$ .

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Step 1.1

Factor out $\frac{x}{y}$ from $\frac{x (x^{2} - y^{2})}{y {(x - y)}^{2}}$ .

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.2

Split $\frac{x^{2} - y^{2}}{{(x - y)}^{2}}$ and simplify.

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Step 1.2.1

Split the fraction $\frac{x^{2} - y^{2}}{{(x - y)}^{2}}$ into two fractions.

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

Step 1.2.2

Move the negative in front of the fraction.

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

Step 1.3

Rewrite $\frac{x^{2}}{{(x - y)}^{2}}$ as ${(\frac{x}{x - y})}^{2}$ .

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

Step 1.4

Multiply $\frac{x}{x - y}$ by $\frac{\frac{1}{y^{1}}}{\frac{1}{y^{1}}}$ .

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

Step 1.5

Multiply $\frac{x}{x - y}$ by $\frac{\frac{1}{y^{1}}}{\frac{1}{y^{1}}}$ .

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

Step 1.6

Apply the distributive property.

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

Step 1.7

Simplify.

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

Step 1.8

Simplify.

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

Step 1.9

Simplify.

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

Step 1.10

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

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

Step 1.11

Simplify each term.

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Step 1.11.1

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

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

Step 1.11.2

Cancel the common factor of $y$ .

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Step 1.11.2.1

Factor $y$ out of $- y$ .

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

Step 1.11.2.2

Cancel the common factor.

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

Step 1.11.2.3

Rewrite the expression.

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

Step 1.12

Rewrite $\frac{y^{2}}{{(x - y)}^{2}}$ as ${(\frac{y}{x - y})}^{2}$ .

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

Step 1.13

Multiply $\frac{y}{x - y}$ by $\frac{\frac{1}{y^{1}}}{\frac{1}{y^{1}}}$ .

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

Step 1.14

Multiply $\frac{y}{x - y}$ by $\frac{\frac{1}{y^{1}}}{\frac{1}{y^{1}}}$ .

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

Step 1.15

Apply the distributive property.

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

Step 1.16

Simplify.

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

Step 1.17

Simplify.

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

Step 1.18

Cancel the common factor of $y$ .

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Step 1.18.1

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

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

Step 1.18.2

Cancel the common factor.

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

Step 1.18.3

Rewrite the expression.

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

Step 1.19

Simplify each term.

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Step 1.19.1

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

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

Step 1.19.2

Cancel the common factor of $y$ .

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Step 1.19.2.1

Factor $y$ out of $- y$ .

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

Step 1.19.2.2

Cancel the common factor.

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

Step 1.19.2.3

Rewrite the expression.

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

Step 2

Let $V = \frac{x}{y}$ . Substitute $V$ for $\frac{x}{y}$ .

$\frac{d x}{d y} = ({(\frac{V}{V - 1})}^{2} - {(\frac{1}{V - 1})}^{2}) V$

Step 3

Solve $V = \frac{x}{y}$ for $x$ .

$x = V y$

Step 4

Use the product rule to find the derivative of $x = V y$ with respect to $y$ .

$\frac{d x}{d y} = y \frac{d V}{d y} + V$

Step 5

Substitute $y \frac{d V}{d y} + V$ for $\frac{d x}{d y}$ .

$y \frac{d V}{d y} + V = ({(\frac{V}{V - 1})}^{2} - {(\frac{1}{V - 1})}^{2}) V$

Step 6

Solve the substituted differential equation.

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Step 6.1

Separate the variables.

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Step 6.1.1

Solve for $\frac{d V}{d y}$ .

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Step 6.1.1.1

Simplify $({(\frac{V}{V - 1})}^{2} - {(\frac{1}{V - 1})}^{2}) V$ .

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Step 6.1.1.1.1

Rewrite.

$y \frac{d V}{d y} + V = 0 + 0 + ({(\frac{V}{V - 1})}^{2} - {(\frac{1}{V - 1})}^{2}) V$

Step 6.1.1.1.2

Simplify by adding zeros.

$y \frac{d V}{d y} + V = ({(\frac{V}{V - 1})}^{2} - {(\frac{1}{V - 1})}^{2}) V$

Step 6.1.1.1.3

Simplify each term.

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Step 6.1.1.1.3.1

Apply the product rule to $\frac{V}{V - 1}$ .

$y \frac{d V}{d y} + V = (\frac{V^{2}}{{(V - 1)}^{2}} - {(\frac{1}{V - 1})}^{2}) V$

Step 6.1.1.1.3.2

Apply the product rule to $\frac{1}{V - 1}$ .

$y \frac{d V}{d y} + V = (\frac{V^{2}}{{(V - 1)}^{2}} - \frac{1^{2}}{{(V - 1)}^{2}}) V$

Step 6.1.1.1.3.3

One to any power is one.

$y \frac{d V}{d y} + V = (\frac{V^{2}}{{(V - 1)}^{2}} - \frac{1}{{(V - 1)}^{2}}) V$

Step 6.1.1.1.4

Apply the distributive property.

$y \frac{d V}{d y} + V = \frac{V^{2}}{{(V - 1)}^{2}} V - \frac{1}{{(V - 1)}^{2}} V$

Step 6.1.1.1.5

Multiply $\frac{V^{2}}{{(V - 1)}^{2}} V$ .

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Step 6.1.1.1.5.1

Combine $\frac{V^{2}}{{(V - 1)}^{2}}$ and $V$ .

$y \frac{d V}{d y} + V = \frac{V^{2} V}{{(V - 1)}^{2}} - \frac{1}{{(V - 1)}^{2}} V$

Step 6.1.1.1.5.2

Multiply $V^{2}$ by $V$ by adding the exponents.

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Step 6.1.1.1.5.2.1

Multiply $V^{2}$ by $V$ .

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Step 6.1.1.1.5.2.1.1

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

$y \frac{d V}{d y} + V = \frac{V^{2} V^{1}}{{(V - 1)}^{2}} - \frac{1}{{(V - 1)}^{2}} V$

Step 6.1.1.1.5.2.1.2

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

$y \frac{d V}{d y} + V = \frac{V^{2 + 1}}{{(V - 1)}^{2}} - \frac{1}{{(V - 1)}^{2}} V$

Step 6.1.1.1.5.2.2

Add $2$ and $1$ .

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

Step 6.1.1.1.6

Combine $V$ and $\frac{1}{{(V - 1)}^{2}}$ .

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

Step 6.1.1.2

Subtract $V$ from both sides of the equation.

$y \frac{d V}{d y} = \frac{V^{3}}{{(V - 1)}^{2}} - \frac{V}{{(V - 1)}^{2}} - V$

Step 6.1.1.3

Divide each term in $y \frac{d V}{d y} = \frac{V^{3}}{{(V - 1)}^{2}} - \frac{V}{{(V - 1)}^{2}} - V$ by $y$ and simplify.

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Step 6.1.1.3.1

Divide each term in $y \frac{d V}{d y} = \frac{V^{3}}{{(V - 1)}^{2}} - \frac{V}{{(V - 1)}^{2}} - V$ by $y$ .

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

Step 6.1.1.3.2

Simplify the left side.

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Step 6.1.1.3.2.1

Cancel the common factor of $y$ .

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Step 6.1.1.3.2.1.1

Cancel the common factor.

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

Step 6.1.1.3.2.1.2

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

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

Step 6.1.1.3.3

Simplify the right side.

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Step 6.1.1.3.3.1

Simplify each term.

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Step 6.1.1.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.1.1.3.3.1.2

Combine.

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

Step 6.1.1.3.3.1.3

Multiply $V^{3}$ by $1$ .

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

Step 6.1.1.3.3.1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.1.1.3.3.1.5

Multiply $\frac{1}{y}$ by $\frac{V}{{(V - 1)}^{2}}$ .

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

Step 6.1.1.3.3.1.6

Move the negative in front of the fraction.

$\frac{d V}{d y} = \frac{V^{3}}{{(V - 1)}^{2} y} - \frac{V}{y {(V - 1)}^{2}} - \frac{V}{y}$

Step 6.1.1.3.3.2

Reorder factors in $\frac{V^{3}}{{(V - 1)}^{2} y} - \frac{V}{y {(V - 1)}^{2}} - \frac{V}{y}$ .

$\frac{d V}{d y} = \frac{V^{3}}{y {(V - 1)}^{2}} - \frac{V}{y {(V - 1)}^{2}} - \frac{V}{y}$

Step 6.1.2

Factor.

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Step 6.1.2.1

Combine the numerators over the common denominator.

$\frac{d V}{d y} = \frac{V^{3} - V}{y {(V - 1)}^{2}} - \frac{V}{y}$

Step 6.1.2.2

Simplify each term.

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Step 6.1.2.2.1

Simplify the numerator.

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Step 6.1.2.2.1.1

Factor $V$ out of $V^{3} - V$ .

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Step 6.1.2.2.1.1.1

Factor $V$ out of $V^{3}$ .

$\frac{d V}{d y} = \frac{V \cdot V^{2} - V}{y {(V - 1)}^{2}} - \frac{V}{y}$

Step 6.1.2.2.1.1.2

Factor $V$ out of $- V$ .

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

Step 6.1.2.2.1.1.3

Factor $V$ out of $V \cdot V^{2} + V \cdot - 1$ .

$\frac{d V}{d y} = \frac{V (V^{2} - 1)}{y {(V - 1)}^{2}} - \frac{V}{y}$

Step 6.1.2.2.1.2

Rewrite $1$ as $1^{2}$ .

$\frac{d V}{d y} = \frac{V (V^{2} - 1^{2})}{y {(V - 1)}^{2}} - \frac{V}{y}$

Step 6.1.2.2.1.3

Factor.

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Step 6.1.2.2.1.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = V$ and $b = 1$ .

$\frac{d V}{d y} = \frac{V ((V + 1) (V - 1))}{y {(V - 1)}^{2}} - \frac{V}{y}$

Step 6.1.2.2.1.3.2

Remove unnecessary parentheses.

$\frac{d V}{d y} = \frac{V (V + 1) (V - 1)}{y {(V - 1)}^{2}} - \frac{V}{y}$

Step 6.1.2.2.2

Cancel the common factor of $V - 1$ and ${(V - 1)}^{2}$ .

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Step 6.1.2.2.2.1

Factor $V - 1$ out of $V (V + 1) (V - 1)$ .

$\frac{d V}{d y} = \frac{(V - 1) (V (V + 1))}{y {(V - 1)}^{2}} - \frac{V}{y}$

Step 6.1.2.2.2.2

Cancel the common factors.

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Step 6.1.2.2.2.2.1

Factor $V - 1$ out of $y {(V - 1)}^{2}$ .

$\frac{d V}{d y} = \frac{(V - 1) (V (V + 1))}{(V - 1) (y (V - 1))} - \frac{V}{y}$

Step 6.1.2.2.2.2.2

Cancel the common factor.

$\frac{d V}{d y} = \frac{(V - 1) (V (V + 1))}{(V - 1) (y (V - 1))} - \frac{V}{y}$

Step 6.1.2.2.2.2.3

Rewrite the expression.

$\frac{d V}{d y} = \frac{V (V + 1)}{y (V - 1)} - \frac{V}{y}$

Step 6.1.2.3

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

$\frac{d V}{d y} = \frac{V (V + 1)}{y (V - 1)} - \frac{V}{y} \cdot \frac{V - 1}{V - 1}$

Step 6.1.2.4

Multiply $\frac{V}{y}$ by $\frac{V - 1}{V - 1}$ .

$\frac{d V}{d y} = \frac{V (V + 1)}{y (V - 1)} - \frac{V (V - 1)}{y (V - 1)}$

Step 6.1.2.5

Combine the numerators over the common denominator.

$\frac{d V}{d y} = \frac{V (V + 1) - V (V - 1)}{y (V - 1)}$

Step 6.1.2.6

Simplify the numerator.

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Step 6.1.2.6.1

Factor $V$ out of $V (V + 1) - V (V - 1)$ .

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Step 6.1.2.6.1.1

Factor $V$ out of $- V (V - 1)$ .

$\frac{d V}{d y} = \frac{V (V + 1) + V (- (V - 1))}{y (V - 1)}$

Step 6.1.2.6.1.2

Factor $V$ out of $V (V + 1) + V (- (V - 1))$ .

$\frac{d V}{d y} = \frac{V (V + 1 - (V - 1))}{y (V - 1)}$

Step 6.1.2.6.2

Apply the distributive property.

$\frac{d V}{d y} = \frac{V (V + 1 - V - - 1)}{y (V - 1)}$

Step 6.1.2.6.3

Multiply $- 1$ by $- 1$ .

$\frac{d V}{d y} = \frac{V (V + 1 - V + 1)}{y (V - 1)}$

Step 6.1.2.6.4

Subtract $V$ from $V$ .

$\frac{d V}{d y} = \frac{V (0 + 1 + 1)}{y (V - 1)}$

Step 6.1.2.6.5

Add $0$ and $1$ .

$\frac{d V}{d y} = \frac{V (1 + 1)}{y (V - 1)}$

Step 6.1.2.6.6

Add $1$ and $1$ .

$\frac{d V}{d y} = \frac{V \cdot 2}{y (V - 1)}$

Step 6.1.2.7

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

$\frac{d V}{d y} = \frac{2 \cdot V}{y (V - 1)}$

Step 6.1.2.8

Multiply $2$ by $V$ .

$\frac{d V}{d y} = \frac{2 V}{y (V - 1)}$

Step 6.1.3

Regroup factors.

$\frac{d V}{d y} = \frac{2 V}{V - 1} \cdot \frac{1}{y}$

Step 6.1.4

Multiply both sides by $\frac{V - 1}{2 V}$ .

$\frac{V - 1}{2 V} \frac{d V}{d y} = \frac{V - 1}{2 V} (\frac{2 V}{V - 1} \cdot \frac{1}{y})$

Step 6.1.5

Simplify.

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Step 6.1.5.1

Multiply $\frac{2 V}{V - 1}$ by $\frac{1}{y}$ .

$\frac{V - 1}{2 V} \frac{d V}{d y} = \frac{V - 1}{2 V} \cdot \frac{2 V}{(V - 1) y}$

Step 6.1.5.2

Cancel the common factor of $V - 1$ .

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Step 6.1.5.2.1

Factor $V - 1$ out of $(V - 1) y$ .

$\frac{V - 1}{2 V} \frac{d V}{d y} = \frac{V - 1}{2 V} \cdot \frac{2 V}{(V - 1) (y)}$

Step 6.1.5.2.2

Cancel the common factor.

$\frac{V - 1}{2 V} \frac{d V}{d y} = \frac{V - 1}{2 V} \cdot \frac{2 V}{(V - 1) y}$

Step 6.1.5.2.3

Rewrite the expression.

$\frac{V - 1}{2 V} \frac{d V}{d y} = \frac{1}{2 V} \cdot \frac{2 V}{y}$

Step 6.1.5.3

Cancel the common factor of $2 V$ .

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Step 6.1.5.3.1

Cancel the common factor.

$\frac{V - 1}{2 V} \frac{d V}{d y} = \frac{1}{2 V} \cdot \frac{2 V}{y}$

Step 6.1.5.3.2

Rewrite the expression.

$\frac{V - 1}{2 V} \frac{d V}{d y} = \frac{1}{y}$

Step 6.1.6

Rewrite the equation.

$\frac{V - 1}{2 V} d V = \frac{1}{y} d y$

Step 6.2

Integrate both sides.

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Step 6.2.1

Set up an integral on each side.

$\int \frac{V - 1}{2 V} d V = \int \frac{1}{y} d y$

Step 6.2.2

Integrate the left side.

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Step 6.2.2.1

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

$\frac{1}{2} \int \frac{V - 1}{V} d V = \int \frac{1}{y} d y$

Step 6.2.2.2

Divide $V - 1$ by $V$ .

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Step 6.2.2.2.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$V$

$0$

$V$

$1$

Step 6.2.2.2.2

Divide the highest order term in the dividend $V$ by the highest order term in divisor $V$ .

					$1$
	$V$	+	$0$		$V$	-	$1$

Step 6.2.2.2.3

Multiply the new quotient term by the divisor.

				$1$
$V$	+	$0$		$V$	-	$1$
			+	$V$	+	$0$

Step 6.2.2.2.4

The expression needs to be subtracted from the dividend, so change all the signs in $V + 0$

				$1$
$V$	+	$0$		$V$	-	$1$
			-	$V$	-	$0$

Step 6.2.2.2.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$1$
$V$	+	$0$		$V$	-	$1$
			-	$V$	-	$0$
					-	$1$

Step 6.2.2.2.6

The final answer is the quotient plus the remainder over the divisor.

$\frac{1}{2} \int 1 - \frac{1}{V} d V = \int \frac{1}{y} d y$

Step 6.2.2.3

Split the single integral into multiple integrals.

$\frac{1}{2} (\int d V + \int - \frac{1}{V} d V) = \int \frac{1}{y} d y$

Step 6.2.2.4

Apply the constant rule.

$\frac{1}{2} (V + C_{1} + \int - \frac{1}{V} d V) = \int \frac{1}{y} d y$

Step 6.2.2.5

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

$\frac{1}{2} (V + C_{1} - \int \frac{1}{V} d V) = \int \frac{1}{y} d y$

Step 6.2.2.6

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

$\frac{1}{2} (V + C_{1} - (\ln (| V |) + C_{2})) = \int \frac{1}{y} d y$

Step 6.2.2.7

Simplify.

$\frac{1}{2} (V - \ln (| V |)) + C_{3} = \int \frac{1}{y} d y$

Step 6.2.3

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

$\frac{1}{2} (V - \ln (| V |)) + C_{3} = \ln (| y |) + C_{4}$

Step 6.2.4

Group the constant of integration on the right side as $C$ .

$\frac{1}{2} (V - \ln (| V |)) = \ln (| y |) + C$

Step 7

Substitute $\frac{x}{y}$ for $V$ .

$\frac{1}{2} (\frac{x}{y} - \ln (| \frac{x}{y} |)) = \ln (| y |) + C$

Step 8

Solve $\frac{1}{2} (\frac{x}{y} - \ln (| \frac{x}{y} |)) = \ln (| y |) + C$ for $x$ .

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Step 8.1

Simplify the left side.

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Step 8.1.1

Simplify $\frac{1}{2} \cdot (\frac{x}{y} - \ln (| \frac{x}{y} |))$ .

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Step 8.1.1.1

Apply the distributive property.

$\frac{1}{2} \cdot \frac{x}{y} + \frac{1}{2} (- \ln (| \frac{x}{y} |)) = \ln (| y |) + C$

Step 8.1.1.2

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

$\frac{x}{2 y} + \frac{1}{2} (- \ln (| \frac{x}{y} |)) = \ln (| y |) + C$

Step 8.1.1.3

Combine $\frac{1}{2}$ and $\ln (| \frac{x}{y} |)$ .

$\frac{x}{2 y} - \frac{\ln (| \frac{x}{y} |)}{2} = \ln (| y |) + C$

Step 8.2

Multiply each term in $\frac{x}{2 y} - \frac{\ln (| \frac{x}{y} |)}{2} = \ln (| y |) + C$ by $2 y$ to eliminate the fractions.

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Step 8.2.1

Multiply each term in $\frac{x}{2 y} - \frac{\ln (| \frac{x}{y} |)}{2} = \ln (| y |) + C$ by $2 y$ .

$\frac{x}{2 y} (2 y) - \frac{\ln (| \frac{x}{y} |)}{2} (2 y) = \ln (| y |) (2 y) + C (2 y)$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Simplify each term.

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Step 8.2.2.1.1

Rewrite using the commutative property of multiplication.

$2 \frac{x}{2 y} y - \frac{\ln (| \frac{x}{y} |)}{2} (2 y) = \ln (| y |) (2 y) + C (2 y)$

Step 8.2.2.1.2

Cancel the common factor of $2$ .

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Step 8.2.2.1.2.1

Factor $2$ out of $2 y$ .

$2 \frac{x}{2 (y)} y - \frac{\ln (| \frac{x}{y} |)}{2} (2 y) = \ln (| y |) (2 y) + C (2 y)$

Step 8.2.2.1.2.2

Cancel the common factor.

$2 \frac{x}{2 y} y - \frac{\ln (| \frac{x}{y} |)}{2} (2 y) = \ln (| y |) (2 y) + C (2 y)$

Step 8.2.2.1.2.3

Rewrite the expression.

$\frac{x}{y} y - \frac{\ln (| \frac{x}{y} |)}{2} (2 y) = \ln (| y |) (2 y) + C (2 y)$

Step 8.2.2.1.3

Cancel the common factor of $y$ .

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Step 8.2.2.1.3.1

Cancel the common factor.

$\frac{x}{y} y - \frac{\ln (| \frac{x}{y} |)}{2} (2 y) = \ln (| y |) (2 y) + C (2 y)$

Step 8.2.2.1.3.2

Rewrite the expression.

$x - \frac{\ln (| \frac{x}{y} |)}{2} (2 y) = \ln (| y |) (2 y) + C (2 y)$

Step 8.2.2.1.4

Cancel the common factor of $2$ .

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Step 8.2.2.1.4.1

Move the leading negative in $- \frac{\ln (| \frac{x}{y} |)}{2}$ into the numerator.

$x + \frac{- \ln (| \frac{x}{y} |)}{2} (2 y) = \ln (| y |) (2 y) + C (2 y)$

Step 8.2.2.1.4.2

Factor $2$ out of $2 y$ .

$x + \frac{- \ln (| \frac{x}{y} |)}{2} (2 (y)) = \ln (| y |) (2 y) + C (2 y)$

Step 8.2.2.1.4.3

Cancel the common factor.

$x + \frac{- \ln (| \frac{x}{y} |)}{2} (2 y) = \ln (| y |) (2 y) + C (2 y)$

Step 8.2.2.1.4.4

Rewrite the expression.

$x - \ln (| \frac{x}{y} |) y = \ln (| y |) (2 y) + C (2 y)$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Simplify each term.

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Step 8.2.3.1.1

Rewrite using the commutative property of multiplication.

$x - \ln (| \frac{x}{y} |) y = 2 \ln (| y |) y + C (2 y)$

Step 8.2.3.1.2

Rewrite using the commutative property of multiplication.

$x - \ln (| \frac{x}{y} |) y = 2 \ln (| y |) y + 2 C y$

Step 8.3

Move all the terms containing a logarithm to the left side of the equation.

$- \ln (| \frac{x}{y} |) y - 2 \ln (| y |) y = - x + 2 C y$

Step 8.4

Reorder factors in $- \ln (| \frac{x}{y} |) y - 2 \ln (| y |) y$ .

$- y \ln (| \frac{x}{y} |) - 2 y \ln (| y |) = - x + 2 C y$

Step 8.5

Simplify the left side.

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Step 8.5.1

Simplify each term.

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Step 8.5.1.1

Simplify $- 2 \ln (| y |)$ by moving $2$ inside the logarithm.

$- y \ln (| \frac{x}{y} |) + y (- \ln ({| y |}^{2})) = - x + 2 C y$

Step 8.5.1.2

Rewrite using the commutative property of multiplication.

$- y \ln (| \frac{x}{y} |) - y \ln ({| y |}^{2}) = - x + 2 C y$

Step 8.5.1.3

Remove the absolute value in ${| y |}^{2}$ because exponentiations with even powers are always positive.

$- y \ln (| \frac{x}{y} |) - y \ln (y^{2}) = - x + 2 C y$

Step 8.6

Subtract $2 C y$ from both sides of the equation.

$- y \ln (| \frac{x}{y} |) - y \ln (y^{2}) - 2 C y = - x$

Step 8.7

Factor $y$ out of $- y \ln (| \frac{x}{y} |) - y \ln (y^{2}) - 2 C y$ .

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Step 8.7.1

Factor $y$ out of $- y \ln (| \frac{x}{y} |)$ .

$y (- 1 \ln (| \frac{x}{y} |)) - y \ln (y^{2}) - 2 C y = - x$

Step 8.7.2

Factor $y$ out of $- y \ln (y^{2})$ .

$y (- 1 \ln (| \frac{x}{y} |)) + y (- 1 \ln (y^{2})) - 2 C y = - x$

Step 8.7.3

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

$y (- 1 \ln (| \frac{x}{y} |)) + y (- 1 \ln (y^{2})) + y (- 2 C) = - x$

Step 8.7.4

Factor $y$ out of $y (- 1 \ln (| \frac{x}{y} |)) + y (- 1 \ln (y^{2}))$ .

$y (- 1 \ln (| \frac{x}{y} |) - 1 \ln (y^{2})) + y (- 2 C) = - x$

Step 8.7.5

Factor $y$ out of $y (- 1 \ln (| \frac{x}{y} |) - 1 \ln (y^{2})) + y (- 2 C)$ .

$y (- 1 \ln (| \frac{x}{y} |) - 1 \ln (y^{2}) - 2 C) = - x$

Step 8.8

Rewrite $- 1 \ln (| \frac{x}{y} |)$ as $- \ln (| \frac{x}{y} |)$ .

$y (- \ln (| \frac{x}{y} |) - 1 \ln (y^{2}) - 2 C) = - x$

Step 8.9

Rewrite $- 1 \ln (y^{2})$ as $- \ln (y^{2})$ .

$y (- \ln (| \frac{x}{y} |) - \ln (y^{2}) - 2 C) = - x$

Step 8.10

Divide each term in $y (- \ln (| \frac{x}{y} |) - \ln (y^{2}) - 2 C) = - x$ by $- \ln (| \frac{x}{y} |) - \ln (y^{2}) - 2 C$ and simplify.

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Step 8.10.1

Divide each term in $y (- \ln (| \frac{x}{y} |) - \ln (y^{2}) - 2 C) = - x$ by $- \ln (| \frac{x}{y} |) - \ln (y^{2}) - 2 C$ .

$\frac{y (- \ln (| \frac{x}{y} |) - \ln (y^{2}) - 2 C)}{- \ln (| \frac{x}{y} |) - \ln (y^{2}) - 2 C} = \frac{- x}{- \ln (| \frac{x}{y} |) - \ln (y^{2}) - 2 C}$

Step 8.10.2

Simplify the left side.

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Step 8.10.2.1

Cancel the common factor of $- \ln (| \frac{x}{y} |) - \ln (y^{2}) - 2 C$ .

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Step 8.10.2.1.1

Cancel the common factor.

$\frac{y (- \ln (| \frac{x}{y} |) - \ln (y^{2}) - 2 C)}{- \ln (| \frac{x}{y} |) - \ln (y^{2}) - 2 C} = \frac{- x}{- \ln (| \frac{x}{y} |) - \ln (y^{2}) - 2 C}$

Step 8.10.2.1.2

Divide $y$ by $1$ .

$y = \frac{- x}{- \ln (| \frac{x}{y} |) - \ln (y^{2}) - 2 C}$

Step 8.10.3

Simplify the right side.

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Step 8.10.3.1

Move the negative in front of the fraction.

$y = - \frac{x}{- \ln (| \frac{x}{y} |) - \ln (y^{2}) - 2 C}$

Step 8.10.3.2

Factor $- 1$ out of $- \ln (| \frac{x}{y} |)$ .

$y = - \frac{x}{- \ln (| \frac{x}{y} |) - \ln (y^{2}) - 2 C}$

Step 8.10.3.3

Factor $- 1$ out of $- \ln (y^{2})$ .

$y = - \frac{x}{- \ln (| \frac{x}{y} |) - \ln (y^{2}) - 2 C}$

Step 8.10.3.4

Factor $- 1$ out of $- \ln (| \frac{x}{y} |) - \ln (y^{2})$ .

$y = - \frac{x}{- (\ln (| \frac{x}{y} |) + \ln (y^{2})) - 2 C}$

Step 8.10.3.5

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

$y = - \frac{x}{- (\ln (| \frac{x}{y} |) + \ln (y^{2})) - (2 C)}$

Step 8.10.3.6

Factor $- 1$ out of $- (\ln (| \frac{x}{y} |) + \ln (y^{2})) - (2 C)$ .

$y = - \frac{x}{- (\ln (| \frac{x}{y} |) + \ln (y^{2}) + 2 C)}$

Step 8.10.3.7

Simplify the expression.

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Step 8.10.3.7.1

Rewrite $- (\ln (| \frac{x}{y} |) + \ln (y^{2}) + 2 C)$ as $- 1 (\ln (| \frac{x}{y} |) + \ln (y^{2}) + 2 C)$ .

$y = - \frac{x}{- 1 (\ln (| \frac{x}{y} |) + \ln (y^{2}) + 2 C)}$

Step 8.10.3.7.2

Move the negative in front of the fraction.

$y = - - \frac{x}{\ln (| \frac{x}{y} |) + \ln (y^{2}) + 2 C}$

Step 8.10.3.7.3

Multiply $- 1$ by $- 1$ .

$y = 1 \frac{x}{\ln (| \frac{x}{y} |) + \ln (y^{2}) + 2 C}$

Step 8.10.3.7.4

Multiply $\frac{x}{\ln (| \frac{x}{y} |) + \ln (y^{2}) + 2 C}$ by $1$ .

$y = \frac{x}{\ln (| \frac{x}{y} |) + \ln (y^{2}) + 2 C}$

Step 8.11

Rewrite the equation as $\frac{x}{\ln (| \frac{x}{y} |) + \ln (y^{2}) + 2 C} = y$ .

$\frac{x}{\ln (| \frac{x}{y} |) + \ln (y^{2}) + 2 C} = y$

Step 8.12

Multiply both sides by $\ln (| \frac{x}{y} |) + \ln (y^{2}) + 2 C$ .

$\frac{x}{\ln (| \frac{x}{y} |) + \ln (y^{2}) + 2 C} (\ln (| \frac{x}{y} |) + \ln (y^{2}) + 2 C) = y (\ln (| \frac{x}{y} |) + \ln (y^{2}) + 2 C)$

Step 8.13

Simplify.

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Step 8.13.1

Simplify the left side.

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Step 8.13.1.1

Cancel the common factor of $\ln (| \frac{x}{y} |) + \ln (y^{2}) + 2 C$ .

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Step 8.13.1.1.1

Cancel the common factor.

$\frac{x}{\ln (| \frac{x}{y} |) + \ln (y^{2}) + 2 C} (\ln (| \frac{x}{y} |) + \ln (y^{2}) + 2 C) = y (\ln (| \frac{x}{y} |) + \ln (y^{2}) + 2 C)$

Step 8.13.1.1.2

Rewrite the expression.

$x = y (\ln (| \frac{x}{y} |) + \ln (y^{2}) + 2 C)$

Step 8.13.2

Simplify the right side.

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Step 8.13.2.1

Simplify $y (\ln (| \frac{x}{y} |) + \ln (y^{2}) + 2 C)$ .

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Step 8.13.2.1.1

Apply the distributive property.

$x = y \ln (| \frac{x}{y} |) + y \ln (y^{2}) + y (2 C)$

Step 8.13.2.1.2

Simplify the expression.

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Step 8.13.2.1.2.1

Rewrite using the commutative property of multiplication.

$x = y \ln (| \frac{x}{y} |) + y \ln (y^{2}) + 2 y C$

Step 8.13.2.1.2.2

Move $y$ .

$x = y \ln (| \frac{x}{y} |) + y \ln (y^{2}) + 2 C y$

Step 8.13.2.1.2.3

Move $y \ln (| \frac{x}{y} |)$ .

$x = y \ln (y^{2}) + 2 C y + y \ln (| \frac{x}{y} |)$

Step 8.13.2.1.2.4

Reorder $y \ln (y^{2})$ and $2 C y$ .

$x = 2 C y + y \ln (y^{2}) + y \ln (| \frac{x}{y} |)$

Step 8.14

Move all the terms containing a logarithm to the left side of the equation.

$- y \ln (y^{2}) - y \ln (| \frac{x}{y} |) = - x + 2 C y$

Step 8.15

Subtract $2 C y$ from both sides of the equation.

$- y \ln (y^{2}) - y \ln (| \frac{x}{y} |) - 2 C y = - x$

Step 8.16

Factor $y$ out of $- y \ln (y^{2}) - y \ln (| \frac{x}{y} |) - 2 C y$ .

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Step 8.16.1

Factor $y$ out of $- y \ln (y^{2})$ .

$y (- 1 \ln (y^{2})) - y \ln (| \frac{x}{y} |) - 2 C y = - x$

Step 8.16.2

Factor $y$ out of $- y \ln (| \frac{x}{y} |)$ .

$y (- 1 \ln (y^{2})) + y (- 1 \ln (| \frac{x}{y} |)) - 2 C y = - x$

Step 8.16.3

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

$y (- 1 \ln (y^{2})) + y (- 1 \ln (| \frac{x}{y} |)) + y (- 2 C) = - x$

Step 8.16.4

Factor $y$ out of $y (- 1 \ln (y^{2})) + y (- 1 \ln (| \frac{x}{y} |))$ .

$y (- 1 \ln (y^{2}) - 1 \ln (| \frac{x}{y} |)) + y (- 2 C) = - x$

Step 8.16.5

Factor $y$ out of $y (- 1 \ln (y^{2}) - 1 \ln (| \frac{x}{y} |)) + y (- 2 C)$ .

$y (- 1 \ln (y^{2}) - 1 \ln (| \frac{x}{y} |) - 2 C) = - x$

Step 8.17

Rewrite $- 1 \ln (y^{2})$ as $- \ln (y^{2})$ .

$y (- \ln (y^{2}) - 1 \ln (| \frac{x}{y} |) - 2 C) = - x$

Step 8.18

Rewrite $- 1 \ln (| \frac{x}{y} |)$ as $- \ln (| \frac{x}{y} |)$ .

$y (- \ln (y^{2}) - \ln (| \frac{x}{y} |) - 2 C) = - x$

Step 8.19

Divide each term in $y (- \ln (y^{2}) - \ln (| \frac{x}{y} |) - 2 C) = - x$ by $- \ln (y^{2}) - \ln (| \frac{x}{y} |) - 2 C$ and simplify.

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Step 8.19.1

Divide each term in $y (- \ln (y^{2}) - \ln (| \frac{x}{y} |) - 2 C) = - x$ by $- \ln (y^{2}) - \ln (| \frac{x}{y} |) - 2 C$ .

$\frac{y (- \ln (y^{2}) - \ln (| \frac{x}{y} |) - 2 C)}{- \ln (y^{2}) - \ln (| \frac{x}{y} |) - 2 C} = \frac{- x}{- \ln (y^{2}) - \ln (| \frac{x}{y} |) - 2 C}$

Step 8.19.2

Simplify the left side.

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Step 8.19.2.1

Cancel the common factor of $- \ln (y^{2}) - \ln (| \frac{x}{y} |) - 2 C$ .

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Step 8.19.2.1.1

Cancel the common factor.

$\frac{y (- \ln (y^{2}) - \ln (| \frac{x}{y} |) - 2 C)}{- \ln (y^{2}) - \ln (| \frac{x}{y} |) - 2 C} = \frac{- x}{- \ln (y^{2}) - \ln (| \frac{x}{y} |) - 2 C}$

Step 8.19.2.1.2

Divide $y$ by $1$ .

$y = \frac{- x}{- \ln (y^{2}) - \ln (| \frac{x}{y} |) - 2 C}$

Step 8.19.3

Simplify the right side.

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Step 8.19.3.1

Move the negative in front of the fraction.

$y = - \frac{x}{- \ln (y^{2}) - \ln (| \frac{x}{y} |) - 2 C}$

Step 8.19.3.2

Factor $- 1$ out of $- \ln (y^{2}) - \ln (| \frac{x}{y} |)$ .

$y = - \frac{x}{- (\ln (y^{2}) + \ln (| \frac{x}{y} |)) - 2 C}$

Step 8.19.3.3

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

$y = - \frac{x}{- (\ln (y^{2}) + \ln (| \frac{x}{y} |)) - (2 C)}$

Step 8.19.3.4

Factor $- 1$ out of $- (\ln (y^{2}) + \ln (| \frac{x}{y} |)) - (2 C)$ .

$y = - \frac{x}{- (\ln (y^{2}) + \ln (| \frac{x}{y} |) + 2 C)}$

Step 8.19.3.5

Simplify the expression.

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Step 8.19.3.5.1

Rewrite $- (\ln (y^{2}) + \ln (| \frac{x}{y} |) + 2 C)$ as $- 1 (\ln (y^{2}) + \ln (| \frac{x}{y} |) + 2 C)$ .

$y = - \frac{x}{- 1 (\ln (y^{2}) + \ln (| \frac{x}{y} |) + 2 C)}$

Step 8.19.3.5.2

Move the negative in front of the fraction.

$y = - - \frac{x}{\ln (y^{2}) + \ln (| \frac{x}{y} |) + 2 C}$

Step 8.19.3.5.3

Multiply $- 1$ by $- 1$ .

$y = 1 \frac{x}{\ln (y^{2}) + \ln (| \frac{x}{y} |) + 2 C}$

Step 8.19.3.5.4

Multiply $\frac{x}{\ln (y^{2}) + \ln (| \frac{x}{y} |) + 2 C}$ by $1$ .

$y = \frac{x}{\ln (y^{2}) + \ln (| \frac{x}{y} |) + 2 C}$