Calculus Examples

$2 x y y' = y^{2} - x^{2}$

Step 1

Rewrite the differential equation.

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

Step 2

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

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $2$ .

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Step 2.1.2.1.1

Cancel the common factor.

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

Step 2.1.2.1.2

Rewrite the expression.

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

Step 2.1.2.2

Cancel the common factor of $x$ .

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Step 2.1.2.2.1

Cancel the common factor.

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

Step 2.1.2.2.2

Rewrite the expression.

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

Step 2.1.2.3

Cancel the common factor of $y$ .

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Step 2.1.2.3.1

Cancel the common factor.

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

Step 2.1.2.3.2

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

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

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Cancel the common factor of $y^{2}$ and $y$ .

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Step 2.1.3.1.1.1

Factor $y$ out of $y^{2}$ .

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

Step 2.1.3.1.1.2

Cancel the common factors.

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Step 2.1.3.1.1.2.1

Factor $y$ out of $2 x y$ .

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

Step 2.1.3.1.1.2.2

Cancel the common factor.

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

Step 2.1.3.1.1.2.3

Rewrite the expression.

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

Step 2.1.3.1.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 2.1.3.1.2.1

Factor $x$ out of $- x^{2}$ .

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

Step 2.1.3.1.2.2

Cancel the common factors.

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Step 2.1.3.1.2.2.1

Factor $x$ out of $2 x y$ .

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

Step 2.1.3.1.2.2.2

Cancel the common factor.

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

Step 2.1.3.1.2.2.3

Rewrite the expression.

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

Step 2.1.3.1.3

Move the negative in front of the fraction.

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

Rewrite the differential equation as $f (\frac{y}{x})$ .

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Step 2.3.1

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

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Step 2.3.1.1

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

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

Step 2.3.1.2

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

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

Step 2.3.2

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

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

Step 3

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

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

Step 4

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

$y = V x$

Step 5

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 6

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

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

Step 7

Solve the substituted differential equation.

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Step 7.1

Separate the variables.

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Step 7.1.1

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

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Step 7.1.1.1

Simplify each term.

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Step 7.1.1.1.1

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

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

Step 7.1.1.1.2

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

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

Step 7.1.1.1.3

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

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

Step 7.1.1.1.4

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

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

Step 7.1.1.2

Subtract $V$ from both sides of the equation.

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

Step 7.1.1.3

Divide each term in $x \frac{d V}{d x} = \frac{V}{2} - \frac{1}{2 V} - V$ by $x$ and simplify.

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Step 7.1.1.3.1

Divide each term in $x \frac{d V}{d x} = \frac{V}{2} - \frac{1}{2 V} - V$ by $x$ .

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

Step 7.1.1.3.2

Simplify the left side.

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Step 7.1.1.3.2.1

Cancel the common factor of $x$ .

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Step 7.1.1.3.2.1.1

Cancel the common factor.

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

Step 7.1.1.3.2.1.2

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

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

Step 7.1.1.3.3

Simplify the right side.

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Step 7.1.1.3.3.1

Combine the numerators over the common denominator.

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

Step 7.1.1.3.3.2

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

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

Step 7.1.1.3.3.3

Simplify terms.

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Step 7.1.1.3.3.3.1

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

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

Step 7.1.1.3.3.3.2

Combine the numerators over the common denominator.

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

Step 7.1.1.3.3.3.3

Simplify each term.

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Step 7.1.1.3.3.3.3.1

Simplify the numerator.

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Step 7.1.1.3.3.3.3.1.1

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

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Step 7.1.1.3.3.3.3.1.1.1

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

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

Step 7.1.1.3.3.3.3.1.1.2

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

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

Step 7.1.1.3.3.3.3.1.1.3

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

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

Step 7.1.1.3.3.3.3.1.1.4

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

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

Step 7.1.1.3.3.3.3.1.2

Multiply $- 1$ by $2$ .

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

Step 7.1.1.3.3.3.3.1.3

Subtract $2$ from $1$ .

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

Step 7.1.1.3.3.3.3.2

Move $- 1$ to the left of $V$ .

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

Step 7.1.1.3.3.3.3.3

Move the negative in front of the fraction.

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

Step 7.1.1.3.3.4

Simplify the numerator.

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Step 7.1.1.3.3.4.1

Factor $- 1$ out of $- \frac{1}{2 V} - \frac{V}{2}$ .

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Step 7.1.1.3.3.4.1.1

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

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

Step 7.1.1.3.3.4.1.2

Factor $- 1$ out of $- \frac{V}{2}$ .

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

Step 7.1.1.3.3.4.1.3

Factor $- 1$ out of $- \frac{1}{2 V}$ .

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

Step 7.1.1.3.3.4.1.4

Factor $- 1$ out of $- (\frac{V}{2}) - (\frac{1}{2 V})$ .

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

Step 7.1.1.3.3.4.2

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

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

Step 7.1.1.3.3.4.3

Multiply $\frac{V}{2}$ by $\frac{V}{V}$ .

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

Step 7.1.1.3.3.4.4

Combine the numerators over the common denominator.

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

Step 7.1.1.3.3.4.5

Multiply $V$ by $V$ .

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

Step 7.1.1.3.3.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.1.1.3.3.6

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

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

Step 7.1.1.3.3.7

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

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

Step 7.1.2

Regroup factors.

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

Step 7.1.3

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

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

Step 7.1.4

Simplify.

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Step 7.1.4.1

Rewrite using the commutative property of multiplication.

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

Step 7.1.4.2

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

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

Step 7.1.4.3

Cancel the common factor of $2 V$ .

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Step 7.1.4.3.1

Move the leading negative in $- \frac{2 V}{V^{2} + 1}$ into the numerator.

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

Step 7.1.4.3.2

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

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

Step 7.1.4.3.3

Factor $2 V$ out of $2 V x$ .

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

Step 7.1.4.3.4

Cancel the common factor.

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

Step 7.1.4.3.5

Rewrite the expression.

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

Step 7.1.4.4

Cancel the common factor of $V^{2} + 1$ .

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Step 7.1.4.4.1

Cancel the common factor.

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

Step 7.1.4.4.2

Rewrite the expression.

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

Step 7.1.5

Rewrite the equation.

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

Step 7.2

Integrate both sides.

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Step 7.2.1

Set up an integral on each side.

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

Step 7.2.2

Integrate the left side.

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Step 7.2.2.1

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

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

Step 7.2.2.2

Let $u = V^{2} + 1$ . Then $d u = 2 V d V$ , so $\frac{1}{2} d u = V d V$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.2.2.2.1

Let $u = V^{2} + 1$ . Find $\frac{d u}{d V}$ .

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Step 7.2.2.2.1.1

Differentiate $V^{2} + 1$ .

$\frac{d}{d V} [V^{2} + 1]$

Step 7.2.2.2.1.2

By the Sum Rule, the derivative of $V^{2} + 1$ with respect to $V$ is $\frac{d}{d V} [V^{2}] + \frac{d}{d V} [1]$ .

$\frac{d}{d V} [V^{2}] + \frac{d}{d V} [1]$

Step 7.2.2.2.1.3

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

$2 V + \frac{d}{d V} [1]$

Step 7.2.2.2.1.4

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

$2 V + 0$

Step 7.2.2.2.1.5

Add $2 V$ and $0$ .

$2 V$

Step 7.2.2.2.2

Rewrite the problem using $u$ and $d u$ .

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

Step 7.2.2.3

Simplify.

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Step 7.2.2.3.1

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

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

Step 7.2.2.3.2

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

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

Step 7.2.2.4

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

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

Step 7.2.2.5

Simplify.

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Step 7.2.2.5.1

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

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

Step 7.2.2.5.2

Cancel the common factor of $2$ .

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Step 7.2.2.5.2.1

Cancel the common factor.

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

Step 7.2.2.5.2.2

Rewrite the expression.

$1 \int \frac{1}{u} d u = \int - \frac{1}{x} d x$

Step 7.2.2.5.3

Multiply $\int \frac{1}{u} d u$ by $1$ .

$\int \frac{1}{u} d u = \int - \frac{1}{x} d x$

Step 7.2.2.6

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

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

Step 7.2.2.7

Replace all occurrences of $u$ with $V^{2} + 1$ .

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

Step 7.2.3

Integrate the right side.

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Step 7.2.3.1

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

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

Step 7.2.3.2

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

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

Step 7.2.3.3

Simplify.

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

Step 7.2.4

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

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

Step 7.3

Solve for $V$ .

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Step 7.3.1

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

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

Step 7.3.2

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

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

Step 7.3.3

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

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

Step 7.3.4

Apply the distributive property.

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

Step 7.3.5

Multiply $x$ by $1$ .

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

Step 7.3.6

To solve for $V$ , rewrite the equation using properties of logarithms.

$e^{\ln (| V^{2} x + x |)} = e^{C}$

Step 7.3.7

Rewrite $\ln (| V^{2} x + x |) = C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{C} = | V^{2} x + x |$

Step 7.3.8

Solve for $V$ .

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Step 7.3.8.1

Rewrite the equation as $| V^{2} x + x | = e^{C}$ .

$| V^{2} x + x | = e^{C}$

Step 7.3.8.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$V^{2} x + x = \pm e^{C}$

Step 7.3.8.3

Subtract $x$ from both sides of the equation.

$V^{2} x = \pm e^{C} - x$

Step 7.3.8.4

Divide each term in $V^{2} x = \pm e^{C} - x$ by $x$ and simplify.

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Step 7.3.8.4.1

Divide each term in $V^{2} x = \pm e^{C} - x$ by $x$ .

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

Step 7.3.8.4.2

Simplify the left side.

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Step 7.3.8.4.2.1

Cancel the common factor of $x$ .

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Step 7.3.8.4.2.1.1

Cancel the common factor.

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

Step 7.3.8.4.2.1.2

Divide $V^{2}$ by $1$ .

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

Step 7.3.8.4.3

Simplify the right side.

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Step 7.3.8.4.3.1

Cancel the common factor of $x$ .

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Step 7.3.8.4.3.1.1

Cancel the common factor.

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

Step 7.3.8.4.3.1.2

Divide $- 1$ by $1$ .

$V^{2} = \frac{\pm e^{C}}{x} - 1$

Step 7.3.8.5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$V = \pm \sqrt{\frac{\pm e^{C}}{x} - 1}$

Step 7.3.8.6

Simplify $\pm \sqrt{\frac{\pm e^{C}}{x} - 1}$ .

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Step 7.3.8.6.1

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

$V = \pm \sqrt{\frac{\pm e^{C}}{x} - 1 \cdot \frac{x}{x}}$

Step 7.3.8.6.2

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

$V = \pm \sqrt{\frac{\pm e^{C}}{x} + \frac{- x}{x}}$

Step 7.3.8.6.3

Combine the numerators over the common denominator.

$V = \pm \sqrt{\frac{\pm e^{C} - x}{x}}$

Step 7.3.8.6.4

Rewrite $\sqrt{\frac{\pm e^{C} - x}{x}}$ as $\frac{\sqrt{\pm e^{C} - x}}{\sqrt{x}}$ .

$V = \pm \frac{\sqrt{\pm e^{C} - x}}{\sqrt{x}}$

Step 7.3.8.6.5

Multiply $\frac{\sqrt{\pm e^{C} - x}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$V = \pm \frac{\sqrt{\pm e^{C} - x}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 7.3.8.6.6

Combine and simplify the denominator.

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Step 7.3.8.6.6.1

Multiply $\frac{\sqrt{\pm e^{C} - x}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$V = \pm \frac{\sqrt{\pm e^{C} - x} \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 7.3.8.6.6.2

Raise $\sqrt{x}$ to the power of $1$ .

$V = \pm \frac{\sqrt{\pm e^{C} - x} \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}}$

Step 7.3.8.6.6.3

Raise $\sqrt{x}$ to the power of $1$ .

$V = \pm \frac{\sqrt{\pm e^{C} - x} \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}}$

Step 7.3.8.6.6.4

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

$V = \pm \frac{\sqrt{\pm e^{C} - x} \sqrt{x}}{{\sqrt{x}}^{1 + 1}}$

Step 7.3.8.6.6.5

Add $1$ and $1$ .

$V = \pm \frac{\sqrt{\pm e^{C} - x} \sqrt{x}}{{\sqrt{x}}^{2}}$

Step 7.3.8.6.6.6

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 7.3.8.6.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$V = \pm \frac{\sqrt{\pm e^{C} - x} \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}}$

Step 7.3.8.6.6.6.2

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

$V = \pm \frac{\sqrt{\pm e^{C} - x} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}}$

Step 7.3.8.6.6.6.3

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

$V = \pm \frac{\sqrt{\pm e^{C} - x} \sqrt{x}}{x^{\frac{2}{2}}}$

Step 7.3.8.6.6.6.4

Cancel the common factor of $2$ .

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Step 7.3.8.6.6.6.4.1

Cancel the common factor.

$V = \pm \frac{\sqrt{\pm e^{C} - x} \sqrt{x}}{x^{\frac{2}{2}}}$

Step 7.3.8.6.6.6.4.2

Rewrite the expression.

$V = \pm \frac{\sqrt{\pm e^{C} - x} \sqrt{x}}{x^{1}}$

Step 7.3.8.6.6.6.5

Simplify.

$V = \pm \frac{\sqrt{\pm e^{C} - x} \sqrt{x}}{x}$

Step 7.3.8.6.7

Combine using the product rule for radicals.

$V = \pm \frac{\sqrt{(\pm e^{C} - x) x}}{x}$

Step 7.3.8.6.8

Reorder factors in $\pm \frac{\sqrt{(\pm e^{C} - x) x}}{x}$ .

$V = \pm \frac{\sqrt{x (\pm e^{C} - x)}}{x}$

Step 7.4

Simplify the constant of integration.

$V = \pm \frac{\sqrt{x (C - x)}}{x}$

Step 8

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

$\frac{y}{x} = \pm \frac{\sqrt{x (C - x)}}{x}$

Step 9

Solve $\frac{y}{x} = \pm \frac{\sqrt{x (C - x)}}{x}$ for $y$ .

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Step 9.1

Multiply both sides by $x$ .

$\frac{y}{x} x = (\pm \frac{\sqrt{x (C - x)}}{x}) x$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $x$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{y}{x} x = (\pm \frac{\sqrt{x (C - x)}}{x}) x$

Step 9.2.1.2

Rewrite the expression.

$y = (\pm \frac{\sqrt{x (C - x)}}{x}) x$