Calculus Examples

$2 y - (2 x - y) \frac{d y}{d x} = 0$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Apply the distributive property.

$2 y + (- (2 x) - - y) \frac{d y}{d x} = 0$

Step 1.1.1.2

Multiply $2$ by $- 1$ .

$2 y + (- 2 x - - y) \frac{d y}{d x} = 0$

Step 1.1.1.3

Multiply $- - y$ .

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Step 1.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 1.1.1.3.2

Multiply $y$ by $1$ .

$2 y + (- 2 x + y) \frac{d y}{d x} = 0$

Step 1.1.1.4

Apply the distributive property.

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

Step 1.1.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

Factor $\frac{d y}{d x}$ out of $y \frac{d y}{d x}$ .

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

Step 1.1.3.3

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

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

Step 1.1.4

Divide each term in $\frac{d y}{d x} (- 2 x + y) = - 2 y$ by $- 2 x + y$ and simplify.

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Step 1.1.4.1

Divide each term in $\frac{d y}{d x} (- 2 x + y) = - 2 y$ by $- 2 x + y$ .

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

Step 1.1.4.2

Simplify the left side.

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Step 1.1.4.2.1

Cancel the common factor of $- 2 x + y$ .

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Step 1.1.4.2.1.1

Cancel the common factor.

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

Step 1.1.4.2.1.2

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

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

Step 1.1.4.3

Simplify the right side.

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Step 1.1.4.3.1

Move the negative in front of the fraction.

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

Step 1.1.4.3.2

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

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

Step 1.1.4.3.3

Factor $- 1$ out of $y$ .

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

Step 1.1.4.3.4

Factor $- 1$ out of $- (2 x) - 1 (- y)$ .

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

Step 1.1.4.3.5

Simplify the expression.

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Step 1.1.4.3.5.1

Rewrite $- (2 x - y)$ as $- 1 (2 x - y)$ .

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

Step 1.1.4.3.5.2

Move the negative in front of the fraction.

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

Step 1.1.4.3.5.3

Multiply $- 1$ by $- 1$ .

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

Step 1.1.4.3.5.4

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Apply the distributive property.

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

Step 1.5

Simplify.

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

Step 1.6

Simplify.

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

Step 1.7

Simplify.

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

Step 1.8

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

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Step 1.8.1

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

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

Step 1.8.2

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

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

Step 1.9

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

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

Step 1.10

Simplify each term.

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Step 1.10.1

Cancel the common factor of $x$ .

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Step 1.10.1.1

Factor $x$ out of $2 x$ .

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

Step 1.10.1.2

Cancel the common factor.

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

Step 1.10.1.3

Rewrite the expression.

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

Step 1.10.2

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

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

Step 1.11

Factor out $\frac{y}{x}$ from $\frac{2 y}{x}$ .

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Step 1.11.1

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

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

Step 1.11.2

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

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

Step 2

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

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

Step 3

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

$y = V x$

Step 4

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

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

Step 5

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

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

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 $\frac{d V}{d x}$ .

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Step 6.1.1.1

Multiply $2$ by $V$ .

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

Step 6.1.1.2

Subtract $V$ from both sides of the equation.

$x \frac{d V}{d x} = \frac{2 V}{2 - V} - V$

Step 6.1.1.3

Divide each term in $x \frac{d V}{d x} = \frac{2 V}{2 - V} - V$ by $x$ and simplify.

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Step 6.1.1.3.1

Divide each term in $x \frac{d V}{d x} = \frac{2 V}{2 - V} - V$ by $x$ .

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

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 $x$ .

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Step 6.1.1.3.2.1.1

Cancel the common factor.

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

Step 6.1.1.3.2.1.2

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

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

Step 6.1.1.3.3

Simplify the right side.

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Step 6.1.1.3.3.1

Combine the numerators over the common denominator.

$\frac{d V}{d x} = \frac{\frac{2 V}{2 - V} - V}{x}$

Step 6.1.1.3.3.2

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

$\frac{d V}{d x} = \frac{\frac{2 V}{2 - V} - V \cdot \frac{2 - V}{2 - V}}{x}$

Step 6.1.1.3.3.3

Simplify terms.

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Step 6.1.1.3.3.3.1

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

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

Step 6.1.1.3.3.3.2

Combine the numerators over the common denominator.

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

Step 6.1.1.3.3.4

Simplify the numerator.

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Step 6.1.1.3.3.4.1

Factor $V$ out of $2 V - V (2 - V)$ .

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Step 6.1.1.3.3.4.1.1

Factor $V$ out of $2 V$ .

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

Step 6.1.1.3.3.4.1.2

Factor $V$ out of $- V (2 - V)$ .

$\frac{d V}{d x} = \frac{\frac{V \cdot 2 + V (- (2 - V))}{2 - V}}{x}$

Step 6.1.1.3.3.4.1.3

Factor $V$ out of $V \cdot 2 + V (- (2 - V))$ .

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

Step 6.1.1.3.3.4.2

Apply the distributive property.

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

Step 6.1.1.3.3.4.3

Multiply $- 1$ by $2$ .

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

Step 6.1.1.3.3.4.4

Multiply $- - V$ .

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Step 6.1.1.3.3.4.4.1

Multiply $- 1$ by $- 1$ .

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

Step 6.1.1.3.3.4.4.2

Multiply $V$ by $1$ .

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

Step 6.1.1.3.3.4.5

Subtract $2$ from $2$ .

$\frac{d V}{d x} = \frac{\frac{V (0 + V)}{2 - V}}{x}$

Step 6.1.1.3.3.4.6

Add $0$ and $V$ .

$\frac{d V}{d x} = \frac{\frac{V \cdot V}{2 - V}}{x}$

Step 6.1.1.3.3.5

Simplify the numerator.

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Step 6.1.1.3.3.5.1

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

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

Step 6.1.1.3.3.5.2

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

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

Step 6.1.1.3.3.5.3

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

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

Step 6.1.1.3.3.5.4

Add $1$ and $1$ .

$\frac{d V}{d x} = \frac{\frac{V^{2}}{2 - V}}{x}$

Step 6.1.1.3.3.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.1.1.3.3.7

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

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

Step 6.1.1.3.3.8

Reorder factors in $\frac{V^{2}}{(2 - V) x}$ .

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

Step 6.1.2

Regroup factors.

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

Step 6.1.3

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

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

Step 6.1.4

Simplify.

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Step 6.1.4.1

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

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

Step 6.1.4.2

Cancel the common factor of $2 - V$ .

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Step 6.1.4.2.1

Factor $2 - V$ out of $(2 - V) x$ .

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

Step 6.1.4.2.2

Cancel the common factor.

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

Step 6.1.4.2.3

Rewrite the expression.

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

Step 6.1.4.3

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

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Step 6.1.4.3.1

Cancel the common factor.

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

Step 6.1.4.3.2

Rewrite the expression.

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

Step 6.1.5

Rewrite the equation.

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

Step 6.2

Integrate both sides.

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Step 6.2.1

Set up an integral on each side.

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

Step 6.2.2

Integrate the left side.

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Step 6.2.2.1

Apply basic rules of exponents.

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Step 6.2.2.1.1

Move $V^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 6.2.2.1.2

Multiply the exponents in ${(V^{2})}^{- 1}$ .

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Step 6.2.2.1.2.1

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

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

Step 6.2.2.1.2.2

Multiply $2$ by $- 1$ .

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

Step 6.2.2.2

Multiply $(2 - V) V^{- 2}$ .

$\int 2 V^{- 2} - V \cdot V^{- 2} d V = \int \frac{1}{x} d x$

Step 6.2.2.3

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

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Step 6.2.2.3.1

Move $V^{- 2}$ .

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

Step 6.2.2.3.2

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

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Step 6.2.2.3.2.1

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

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

Step 6.2.2.3.2.2

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

$\int 2 V^{- 2} - V^{- 2 + 1} d V = \int \frac{1}{x} d x$

Step 6.2.2.3.3

Add $- 2$ and $1$ .

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

Step 6.2.2.4

Split the single integral into multiple integrals.

$\int 2 V^{- 2} d V + \int - V^{- 1} d V = \int \frac{1}{x} d x$

Step 6.2.2.5

Since $2$ is constant with respect to $V$ , move $2$ out of the integral.

$2 \int V^{- 2} d V + \int - V^{- 1} d V = \int \frac{1}{x} d x$

Step 6.2.2.6

By the Power Rule, the integral of $V^{- 2}$ with respect to $V$ is $- V^{- 1}$ .

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

Step 6.2.2.7

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

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

Step 6.2.2.8

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

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

Step 6.2.2.9

Simplify.

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Step 6.2.2.9.1

Simplify.

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

Step 6.2.2.9.2

Simplify.

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Step 6.2.2.9.2.1

Multiply $- 1$ by $2$ .

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

Step 6.2.2.9.2.2

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

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

Step 6.2.2.9.2.3

Move the negative in front of the fraction.

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

Step 6.2.3

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

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

Step 6.2.4

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

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Simplify each term.

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Step 8.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2.2

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

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

Step 8.3

Divide each term in $- \ln (| \frac{y}{x} |) = \frac{2 x}{y} + C + \ln (| x |)$ by $- 1$ and simplify.

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Step 8.3.1

Divide each term in $- \ln (| \frac{y}{x} |) = \frac{2 x}{y} + C + \ln (| x |)$ by $- 1$ .

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

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Dividing two negative values results in a positive value.

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

Step 8.3.2.2

Divide $\ln (| \frac{y}{x} |)$ by $1$ .

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

Step 8.3.3

Simplify the right side.

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Step 8.3.3.1

Simplify each term.

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Step 8.3.3.1.1

Move the negative one from the denominator of $\frac{\frac{2 x}{y}}{- 1}$ .

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

Step 8.3.3.1.2

Rewrite $- 1 \cdot \frac{2 x}{y}$ as $- \frac{2 x}{y}$ .

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

Step 8.3.3.1.3

Move the negative one from the denominator of $\frac{C}{- 1}$ .

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

Step 8.3.3.1.4

Rewrite $- 1 \cdot C$ as $- C$ .

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

Step 8.3.3.1.5

Move the negative one from the denominator of $\frac{\ln (| x |)}{- 1}$ .

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

Step 8.3.3.1.6

Rewrite $- 1 \cdot \ln (| x |)$ as $- \ln (| x |)$ .

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

Step 8.4

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

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

Step 8.5

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 8.6

Multiply $| \frac{y}{x} | | x |$ .

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Step 8.6.1

To multiply absolute values, multiply the terms inside each absolute value.

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

Step 8.6.2

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

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

Step 8.7

Cancel the common factor of $x$ .

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Step 8.7.1

Cancel the common factor.

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

Step 8.7.2

Divide $y$ by $1$ .

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