Calculus Examples

$3 x y \frac{d y}{d x} = 4 x^{2} + 9 y^{2}$

Step 1

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

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Step 1.1

Divide each term in $3 x y \frac{d y}{d x} = 4 x^{2} + 9 y^{2}$ by $3 x y$ and simplify.

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Step 1.1.1

Divide each term in $3 x y \frac{d y}{d x} = 4 x^{2} + 9 y^{2}$ by $3 x y$ .

$\frac{3 x y \frac{d y}{d x}}{3 x y} = \frac{4 x^{2}}{3 x y} + \frac{9 y^{2}}{3 x y}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $3$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{3 x y \frac{d y}{d x}}{3 x y} = \frac{4 x^{2}}{3 x y} + \frac{9 y^{2}}{3 x y}$

Step 1.1.2.1.2

Rewrite the expression.

$\frac{x y \frac{d y}{d x}}{x y} = \frac{4 x^{2}}{3 x y} + \frac{9 y^{2}}{3 x y}$

Step 1.1.2.2

Cancel the common factor of $x$ .

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Step 1.1.2.2.1

Cancel the common factor.

$\frac{x y \frac{d y}{d x}}{x y} = \frac{4 x^{2}}{3 x y} + \frac{9 y^{2}}{3 x y}$

Step 1.1.2.2.2

Rewrite the expression.

$\frac{y \frac{d y}{d x}}{y} = \frac{4 x^{2}}{3 x y} + \frac{9 y^{2}}{3 x y}$

Step 1.1.2.3

Cancel the common factor of $y$ .

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Step 1.1.2.3.1

Cancel the common factor.

$\frac{y \frac{d y}{d x}}{y} = \frac{4 x^{2}}{3 x y} + \frac{9 y^{2}}{3 x y}$

Step 1.1.2.3.2

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

$\frac{d y}{d x} = \frac{4 x^{2}}{3 x y} + \frac{9 y^{2}}{3 x y}$

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

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

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Step 1.1.3.1.1.1

Factor $x$ out of $4 x^{2}$ .

$\frac{d y}{d x} = \frac{x (4 x)}{3 x y} + \frac{9 y^{2}}{3 x y}$

Step 1.1.3.1.1.2

Cancel the common factors.

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Step 1.1.3.1.1.2.1

Factor $x$ out of $3 x y$ .

$\frac{d y}{d x} = \frac{x (4 x)}{x (3 y)} + \frac{9 y^{2}}{3 x y}$

Step 1.1.3.1.1.2.2

Cancel the common factor.

$\frac{d y}{d x} = \frac{x (4 x)}{x (3 y)} + \frac{9 y^{2}}{3 x y}$

Step 1.1.3.1.1.2.3

Rewrite the expression.

$\frac{d y}{d x} = \frac{4 x}{3 y} + \frac{9 y^{2}}{3 x y}$

Step 1.1.3.1.2

Cancel the common factor of $9$ and $3$ .

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Step 1.1.3.1.2.1

Factor $3$ out of $9 y^{2}$ .

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

Step 1.1.3.1.2.2

Cancel the common factors.

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Step 1.1.3.1.2.2.1

Factor $3$ out of $3 x y$ .

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

Step 1.1.3.1.2.2.2

Cancel the common factor.

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

Step 1.1.3.1.2.2.3

Rewrite the expression.

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

Step 1.1.3.1.3

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

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Step 1.1.3.1.3.1

Factor $y$ out of $3 y^{2}$ .

$\frac{d y}{d x} = \frac{4 x}{3 y} + \frac{y (3 y)}{x y}$

Step 1.1.3.1.3.2

Cancel the common factors.

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Step 1.1.3.1.3.2.1

Factor $y$ out of $x y$ .

$\frac{d y}{d x} = \frac{4 x}{3 y} + \frac{y (3 y)}{y x}$

Step 1.1.3.1.3.2.2

Cancel the common factor.

$\frac{d y}{d x} = \frac{4 x}{3 y} + \frac{y (3 y)}{y x}$

Step 1.1.3.1.3.2.3

Rewrite the expression.

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

Step 1.2

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

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Step 1.2.1

Factor out $\frac{x}{y}$ from $\frac{4 x}{3 y}$ .

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Step 1.2.1.1

Factor $\frac{x}{y}$ out of $\frac{4 x}{3 y}$ .

$\frac{d y}{d x} = \frac{x}{y} \cdot \frac{4}{3} + \frac{3 y}{x}$

Step 1.2.1.2

Reorder $\frac{x}{y}$ and $\frac{4}{3}$ .

$\frac{d y}{d x} = \frac{4}{3} \cdot \frac{x}{y} + \frac{3 y}{x}$

Step 1.2.2

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

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

Step 1.3

Factor out $\frac{y}{x}$ from $\frac{3 y}{x}$ .

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Step 1.3.1

Factor $\frac{y}{x}$ out of $\frac{3 y}{x}$ .

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

Step 1.3.2

Reorder $\frac{y}{x}$ and $3$ .

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

Step 2

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

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

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

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

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

Simplify each term.

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Step 6.1.1.1.1

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

$x \frac{d V}{d x} + V = \frac{4}{3} \cdot \frac{1}{V} + 3 \cdot V$

Step 6.1.1.1.2

Multiply $\frac{4}{3}$ by $\frac{1}{V}$ .

$x \frac{d V}{d x} + V = \frac{4}{3 V} + 3 V$

Step 6.1.1.2

Move all terms not containing $\frac{d V}{d x}$ to the right side of the equation.

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Step 6.1.1.2.1

Subtract $V$ from both sides of the equation.

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

Step 6.1.1.2.2

Subtract $V$ from $3 V$ .

$x \frac{d V}{d x} = \frac{4}{3 V} + 2 V$

Step 6.1.1.3

Divide each term in $x \frac{d V}{d x} = \frac{4}{3 V} + 2 V$ by $x$ and simplify.

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Step 6.1.1.3.1

Divide each term in $x \frac{d V}{d x} = \frac{4}{3 V} + 2 V$ by $x$ .

$\frac{x \frac{d V}{d x}}{x} = \frac{\frac{4}{3 V}}{x} + \frac{2 V}{x}$

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.

$\frac{x \frac{d V}{d x}}{x} = \frac{\frac{4}{3 V}}{x} + \frac{2 V}{x}$

Step 6.1.1.3.2.1.2

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

$\frac{d V}{d x} = \frac{\frac{4}{3 V}}{x} + \frac{2 V}{x}$

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.

$\frac{d V}{d x} = \frac{4}{3 V} \cdot \frac{1}{x} + \frac{2 V}{x}$

Step 6.1.1.3.3.1.2

Multiply $\frac{4}{3 V}$ by $\frac{1}{x}$ .

$\frac{d V}{d x} = \frac{4}{3 V x} + \frac{2 V}{x}$

Step 6.1.2

Factor.

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Step 6.1.2.1

To write $\frac{2 V}{x}$ as a fraction with a common denominator, multiply by $\frac{3 V}{3 V}$ .

$\frac{d V}{d x} = \frac{4}{3 V x} + \frac{2 V}{x} \cdot \frac{3 V}{3 V}$

Step 6.1.2.2

Write each expression with a common denominator of $3 V x$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1.2.2.1

Multiply $\frac{2 V}{x}$ by $\frac{3 V}{3 V}$ .

$\frac{d V}{d x} = \frac{4}{3 V x} + \frac{2 V (3 V)}{x (3 V)}$

Step 6.1.2.2.2

Reorder the factors of $x (3 V)$ .

$\frac{d V}{d x} = \frac{4}{3 V x} + \frac{2 V (3 V)}{3 V x}$

Step 6.1.2.3

Combine the numerators over the common denominator.

$\frac{d V}{d x} = \frac{4 + 2 V (3 V)}{3 V x}$

Step 6.1.2.4

Simplify the numerator.

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Step 6.1.2.4.1

Rewrite using the commutative property of multiplication.

$\frac{d V}{d x} = \frac{4 + 2 \cdot 3 V \cdot V}{3 V x}$

Step 6.1.2.4.2

Multiply $V$ by $V$ by adding the exponents.

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Step 6.1.2.4.2.1

Move $V$ .

$\frac{d V}{d x} = \frac{4 + 2 \cdot 3 (V \cdot V)}{3 V x}$

Step 6.1.2.4.2.2

Multiply $V$ by $V$ .

$\frac{d V}{d x} = \frac{4 + 2 \cdot 3 V^{2}}{3 V x}$

Step 6.1.2.4.3

Multiply $2$ by $3$ .

$\frac{d V}{d x} = \frac{4 + 6 V^{2}}{3 V x}$

Step 6.1.2.4.4

Factor $2$ out of $4 + 6 V^{2}$ .

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Step 6.1.2.4.4.1

Factor $2$ out of $4$ .

$\frac{d V}{d x} = \frac{2 (2) + 6 V^{2}}{3 V x}$

Step 6.1.2.4.4.2

Factor $2$ out of $6 V^{2}$ .

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

Step 6.1.2.4.4.3

Factor $2$ out of $2 (2) + 2 (3 V^{2})$ .

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

Step 6.1.3

Regroup factors.

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

Step 6.1.4

Multiply both sides by $\frac{3 V}{2 (2 + 3 V^{2})}$ .

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

Step 6.1.5

Simplify.

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Step 6.1.5.1

Combine.

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

Step 6.1.5.2

Combine.

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

Step 6.1.5.3

Cancel the common factor of $3$ .

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Step 6.1.5.3.1

Cancel the common factor.

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

Step 6.1.5.3.2

Rewrite the expression.

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

Step 6.1.5.4

Cancel the common factor of $V$ .

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Step 6.1.5.4.1

Cancel the common factor.

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

Step 6.1.5.4.2

Rewrite the expression.

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

Step 6.1.5.5

Cancel the common factor of $2$ .

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Step 6.1.5.5.1

Cancel the common factor.

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

Step 6.1.5.5.2

Rewrite the expression.

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

Step 6.1.5.6

Cancel the common factor of $2 + 3 V^{2}$ .

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Step 6.1.5.6.1

Cancel the common factor.

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

Step 6.1.5.6.2

Rewrite the expression.

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

Step 6.1.6

Rewrite the equation.

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

Step 6.2

Integrate both sides.

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Step 6.2.1

Set up an integral on each side.

$\int \frac{3 V}{2 (2 + 3 V^{2})} d V = \int \frac{1}{x} d x$

Step 6.2.2

Integrate the left side.

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Step 6.2.2.1

Since $\frac{3}{2}$ is constant with respect to $V$ , move $\frac{3}{2}$ out of the integral.

$\frac{3}{2} \int \frac{V}{2 + 3 V^{2}} d V = \int \frac{1}{x} d x$

Step 6.2.2.2

Let $u = 2 + 3 V^{2}$ . Then $d u = 6 V d V$ , so $\frac{1}{6} d u = V d V$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.2.2.2.1

Let $u = 2 + 3 V^{2}$ . Find $\frac{d u}{d V}$ .

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Step 6.2.2.2.1.1

Differentiate $2 + 3 V^{2}$ .

$\frac{d}{d V} [2 + 3 V^{2}]$

Step 6.2.2.2.1.2

Differentiate.

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Step 6.2.2.2.1.2.1

By the Sum Rule, the derivative of $2 + 3 V^{2}$ with respect to $V$ is $\frac{d}{d V} [2] + \frac{d}{d V} [3 V^{2}]$ .

$\frac{d}{d V} [2] + \frac{d}{d V} [3 V^{2}]$

Step 6.2.2.2.1.2.2

Since $2$ is constant with respect to $V$ , the derivative of $2$ with respect to $V$ is $0$ .

$0 + \frac{d}{d V} [3 V^{2}]$

Step 6.2.2.2.1.3

Evaluate $\frac{d}{d V} [3 V^{2}]$ .

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Step 6.2.2.2.1.3.1

Since $3$ is constant with respect to $V$ , the derivative of $3 V^{2}$ with respect to $V$ is $3 \frac{d}{d V} [V^{2}]$ .

$0 + 3 \frac{d}{d V} [V^{2}]$

Step 6.2.2.2.1.3.2

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

$0 + 3 (2 V)$

Step 6.2.2.2.1.3.3

Multiply $2$ by $3$ .

$0 + 6 V$

Step 6.2.2.2.1.4

Add $0$ and $6 V$ .

$6 V$

Step 6.2.2.2.2

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

$\frac{3}{2} \int \frac{1}{u} \cdot \frac{1}{6} d u = \int \frac{1}{x} d x$

Step 6.2.2.3

Simplify.

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Step 6.2.2.3.1

Multiply $\frac{1}{u}$ by $\frac{1}{6}$ .

$\frac{3}{2} \int \frac{1}{u \cdot 6} d u = \int \frac{1}{x} d x$

Step 6.2.2.3.2

Move $6$ to the left of $u$ .

$\frac{3}{2} \int \frac{1}{6 u} d u = \int \frac{1}{x} d x$

Step 6.2.2.4

Since $\frac{1}{6}$ is constant with respect to $u$ , move $\frac{1}{6}$ out of the integral.

$\frac{3}{2} (\frac{1}{6} \int \frac{1}{u} d u) = \int \frac{1}{x} d x$

Step 6.2.2.5

Simplify.

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Step 6.2.2.5.1

Multiply $\frac{1}{6}$ by $\frac{3}{2}$ .

$\frac{3}{6 \cdot 2} \int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 6.2.2.5.2

Multiply $6$ by $2$ .

$\frac{3}{12} \int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 6.2.2.5.3

Cancel the common factor of $3$ and $12$ .

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Step 6.2.2.5.3.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{12} \int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 6.2.2.5.3.2

Cancel the common factors.

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Step 6.2.2.5.3.2.1

Factor $3$ out of $12$ .

$\frac{3 \cdot 1}{3 \cdot 4} \int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 6.2.2.5.3.2.2

Cancel the common factor.

$\frac{3 \cdot 1}{3 \cdot 4} \int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 6.2.2.5.3.2.3

Rewrite the expression.

$\frac{1}{4} \int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 6.2.2.6

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

$\frac{1}{4} (\ln (| u |) + C_{1}) = \int \frac{1}{x} d x$

Step 6.2.2.7

Simplify.

$\frac{1}{4} \ln (| u |) + C_{1} = \int \frac{1}{x} d x$

Step 6.2.2.8

Replace all occurrences of $u$ with $2 + 3 V^{2}$ .

$\frac{1}{4} \ln (| 2 + 3 V^{2} |) + C_{1} = \int \frac{1}{x} d x$

Step 6.2.3

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

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

Step 6.2.4

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

$\frac{1}{4} \ln (| 2 + 3 V^{2} |) = \ln (| x |) + C$

Step 6.3

Solve for $V$ .

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Step 6.3.1

Multiply both sides of the equation by $4$ .

$4 (\frac{1}{4} \ln (| 2 + 3 V^{2} |)) = 4 (\ln (| x |) + C)$

Step 6.3.2

Simplify both sides of the equation.

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Step 6.3.2.1

Simplify the left side.

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Step 6.3.2.1.1

Simplify $4 (\frac{1}{4} \ln (| 2 + 3 V^{2} |))$ .

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Step 6.3.2.1.1.1

Combine $\frac{1}{4}$ and $\ln (| 2 + 3 V^{2} |)$ .

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

Step 6.3.2.1.1.2

Cancel the common factor of $4$ .

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Step 6.3.2.1.1.2.1

Cancel the common factor.

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

Step 6.3.2.1.1.2.2

Rewrite the expression.

$\ln (| 2 + 3 V^{2} |) = 4 (\ln (| x |) + C)$

Step 6.3.2.2

Simplify the right side.

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Step 6.3.2.2.1

Apply the distributive property.

$\ln (| 2 + 3 V^{2} |) = 4 \ln (| x |) + 4 C$

Step 6.3.3

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

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

Step 6.3.4

Simplify the left side.

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Step 6.3.4.1

Simplify $\ln (| 2 + 3 V^{2} |) - 4 \ln (| x |)$ .

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Step 6.3.4.1.1

Simplify each term.

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Step 6.3.4.1.1.1

Simplify $- 4 \ln (| x |)$ by moving $4$ inside the logarithm.

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

Step 6.3.4.1.1.2

Remove the absolute value in ${| x |}^{4}$ because exponentiations with even powers are always positive.

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

Step 6.3.4.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

Step 6.3.5

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

$e^{\ln (\frac{| 2 + 3 V^{2} |}{x^{4}})} = e^{4 C}$

Step 6.3.6

Rewrite $\ln (\frac{| 2 + 3 V^{2} |}{x^{4}}) = 4 C$ 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 $b^{y} = x$ .

$e^{4 C} = \frac{| 2 + 3 V^{2} |}{x^{4}}$

Step 6.3.7

Solve for $V$ .

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Step 6.3.7.1

Rewrite the equation as $\frac{| 2 + 3 V^{2} |}{x^{4}} = e^{4 C}$ .

$\frac{| 2 + 3 V^{2} |}{x^{4}} = e^{4 C}$

Step 6.3.7.2

Multiply both sides by $x^{4}$ .

$\frac{| 2 + 3 V^{2} |}{x^{4}} x^{4} = e^{4 C} x^{4}$

Step 6.3.7.3

Simplify.

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Step 6.3.7.3.1

Simplify the left side.

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Step 6.3.7.3.1.1

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

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Step 6.3.7.3.1.1.1

Cancel the common factor.

$\frac{| 2 + 3 V^{2} |}{x^{4}} x^{4} = e^{4 C} x^{4}$

Step 6.3.7.3.1.1.2

Rewrite the expression.

$| 2 + 3 V^{2} | = e^{4 C} x^{4}$

Step 6.3.7.3.2

Simplify the right side.

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Step 6.3.7.3.2.1

Reorder factors in $e^{4 C} x^{4}$ .

$| 2 + 3 V^{2} | = x^{4} e^{4 C}$

Step 6.3.7.4

Solve for $V$ .

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Step 6.3.7.4.1

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

$2 + 3 V^{2} = \pm x^{4} e^{4 C}$

Step 6.3.7.4.2

Subtract $2$ from both sides of the equation.

$3 V^{2} = \pm x^{4} e^{4 C} - 2$

Step 6.3.7.4.3

Divide each term in $3 V^{2} = \pm x^{4} e^{4 C} - 2$ by $3$ and simplify.

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Step 6.3.7.4.3.1

Divide each term in $3 V^{2} = \pm x^{4} e^{4 C} - 2$ by $3$ .

$\frac{3 V^{2}}{3} = \frac{\pm x^{4} e^{4 C}}{3} + \frac{- 2}{3}$

Step 6.3.7.4.3.2

Simplify the left side.

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Step 6.3.7.4.3.2.1

Cancel the common factor of $3$ .

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Step 6.3.7.4.3.2.1.1

Cancel the common factor.

$\frac{3 V^{2}}{3} = \frac{\pm x^{4} e^{4 C}}{3} + \frac{- 2}{3}$

Step 6.3.7.4.3.2.1.2

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

$V^{2} = \frac{\pm x^{4} e^{4 C}}{3} + \frac{- 2}{3}$

Step 6.3.7.4.3.3

Simplify the right side.

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Step 6.3.7.4.3.3.1

Move the negative in front of the fraction.

$V^{2} = \frac{\pm x^{4} e^{4 C}}{3} - \frac{2}{3}$

Step 6.3.7.4.4

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

$V = \pm \sqrt{\frac{\pm x^{4} e^{4 C}}{3} - \frac{2}{3}}$

Step 6.3.7.4.5

Simplify $\pm \sqrt{\frac{\pm x^{4} e^{4 C}}{3} - \frac{2}{3}}$ .

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Step 6.3.7.4.5.1

Combine the numerators over the common denominator.

$V = \pm \sqrt{\frac{\pm x^{4} e^{4 C} - 2}{3}}$

Step 6.3.7.4.5.2

Rewrite $\sqrt{\frac{\pm x^{4} e^{4 C} - 2}{3}}$ as $\frac{\sqrt{\pm x^{4} e^{4 C} - 2}}{\sqrt{3}}$ .

$V = \pm \frac{\sqrt{\pm x^{4} e^{4 C} - 2}}{\sqrt{3}}$

Step 6.3.7.4.5.3

Multiply $\frac{\sqrt{\pm x^{4} e^{4 C} - 2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$V = \pm \frac{\sqrt{\pm x^{4} e^{4 C} - 2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 6.3.7.4.5.4

Combine and simplify the denominator.

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Step 6.3.7.4.5.4.1

Multiply $\frac{\sqrt{\pm x^{4} e^{4 C} - 2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$V = \pm \frac{\sqrt{\pm x^{4} e^{4 C} - 2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 6.3.7.4.5.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$V = \pm \frac{\sqrt{\pm x^{4} e^{4 C} - 2} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 6.3.7.4.5.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$V = \pm \frac{\sqrt{\pm x^{4} e^{4 C} - 2} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 6.3.7.4.5.4.4

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

$V = \pm \frac{\sqrt{\pm x^{4} e^{4 C} - 2} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 6.3.7.4.5.4.5

Add $1$ and $1$ .

$V = \pm \frac{\sqrt{\pm x^{4} e^{4 C} - 2} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 6.3.7.4.5.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 6.3.7.4.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$V = \pm \frac{\sqrt{\pm x^{4} e^{4 C} - 2} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 6.3.7.4.5.4.6.2

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

$V = \pm \frac{\sqrt{\pm x^{4} e^{4 C} - 2} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 6.3.7.4.5.4.6.3

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

$V = \pm \frac{\sqrt{\pm x^{4} e^{4 C} - 2} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 6.3.7.4.5.4.6.4

Cancel the common factor of $2$ .

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Step 6.3.7.4.5.4.6.4.1

Cancel the common factor.

$V = \pm \frac{\sqrt{\pm x^{4} e^{4 C} - 2} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 6.3.7.4.5.4.6.4.2

Rewrite the expression.

$V = \pm \frac{\sqrt{\pm x^{4} e^{4 C} - 2} \sqrt{3}}{3^{1}}$

Step 6.3.7.4.5.4.6.5

Evaluate the exponent.

$V = \pm \frac{\sqrt{\pm x^{4} e^{4 C} - 2} \sqrt{3}}{3}$

Step 6.3.7.4.5.5

Combine using the product rule for radicals.

$V = \pm \frac{\sqrt{(\pm x^{4} e^{4 C} - 2) \cdot 3}}{3}$

Step 6.3.7.4.5.6

Reorder factors in $\pm \frac{\sqrt{(\pm x^{4} e^{4 C} - 2) \cdot 3}}{3}$ .

$V = \pm \frac{\sqrt{3 (\pm x^{4} e^{4 C} - 2)}}{3}$

Step 6.4

Group the constant terms together.

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Step 6.4.1

Simplify the constant of integration.

$V = \pm \frac{\sqrt{3 (\pm x^{4} C - 2)}}{3}$

Step 6.4.2

Combine constants with the plus or minus.

$V = \pm \frac{\sqrt{3 (C x^{4} - 2)}}{3}$

Step 7

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

$\frac{y}{x} = \pm \frac{\sqrt{3 (C x^{4} - 2)}}{3}$

Step 8

Solve $\frac{y}{x} = \pm \frac{\sqrt{3 (C x^{4} - 2)}}{3}$ for $y$ .

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Step 8.1

Multiply both sides by $x$ .

$\frac{y}{x} x = (\pm \frac{\sqrt{3 (C x^{4} - 2)}}{3}) x$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $x$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{y}{x} x = (\pm \frac{\sqrt{3 (C x^{4} - 2)}}{3}) x$

Step 8.2.1.2

Rewrite the expression.

$y = (\pm \frac{\sqrt{3 (C x^{4} - 2)}}{3}) x$