Calculus Examples

$x^{2} + y^{2} + x y y' = 0$

Step 1

Rewrite the differential equation.

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

Step 2

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

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Step 2.1

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

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Step 2.1.1

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

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Step 2.1.1.1

Subtract $x^{2}$ from both sides of the equation.

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

Step 2.1.1.2

Subtract $y^{2}$ from both sides of the equation.

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

Step 2.1.2

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

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Step 2.1.2.1

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

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

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $x$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

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

Step 2.1.2.2.1.2

Rewrite the expression.

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

Step 2.1.2.2.2

Cancel the common factor of $y$ .

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Step 2.1.2.2.2.1

Cancel the common factor.

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

Step 2.1.2.2.2.2

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

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

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

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

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Step 2.1.2.3.1.1.1

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

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

Step 2.1.2.3.1.1.2

Cancel the common factors.

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Step 2.1.2.3.1.1.2.1

Factor $x$ out of $x y$ .

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

Step 2.1.2.3.1.1.2.2

Cancel the common factor.

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

Step 2.1.2.3.1.1.2.3

Rewrite the expression.

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

Step 2.1.2.3.1.2

Move the negative in front of the fraction.

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

Step 2.1.2.3.1.3

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

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Step 2.1.2.3.1.3.1

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

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

Step 2.1.2.3.1.3.2

Cancel the common factors.

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Step 2.1.2.3.1.3.2.1

Factor $y$ out of $x y$ .

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

Step 2.1.2.3.1.3.2.2

Cancel the common factor.

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

Step 2.1.2.3.1.3.2.3

Rewrite the expression.

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

Step 2.1.2.3.1.4

Move the negative in front of the fraction.

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

Step 2.2

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

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

Step 3

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

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

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

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

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

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

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

Step 7.1.1.2

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

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Step 7.1.1.2.1

Subtract $V$ from both sides of the equation.

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

Step 7.1.1.2.2

Subtract $V$ from $- V$ .

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

Step 7.1.1.3

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

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Step 7.1.1.3.1

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

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

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.

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

Step 7.1.1.3.2.1.2

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

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

Step 7.1.1.3.3

Simplify the right side.

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Step 7.1.1.3.3.1

Simplify each term.

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Step 7.1.1.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.1.1.3.3.1.2

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

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

Step 7.1.1.3.3.1.3

Move the negative in front of the fraction.

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

Step 7.1.2

Factor.

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Step 7.1.2.1

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

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Step 7.1.2.1.1

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

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

Step 7.1.2.1.2

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

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

Step 7.1.2.1.3

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

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

Step 7.1.2.1.4

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

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

Step 7.1.2.2

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

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

Step 7.1.2.3

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

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

Step 7.1.2.4

Combine the numerators over the common denominator.

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

Step 7.1.2.5

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

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Step 7.1.2.5.1

Move $V$ .

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

Step 7.1.2.5.2

Multiply $V$ by $V$ .

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

Step 7.1.3

Regroup factors.

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

Step 7.1.4

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

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

Step 7.1.5

Simplify.

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Step 7.1.5.1

Rewrite using the commutative property of multiplication.

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

Step 7.1.5.2

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

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

Step 7.1.5.3

Cancel the common factor of $V$ .

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Step 7.1.5.3.1

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

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

Step 7.1.5.3.2

Factor $V$ out of $- V$ .

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

Step 7.1.5.3.3

Factor $V$ out of $V x$ .

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

Step 7.1.5.3.4

Cancel the common factor.

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

Step 7.1.5.3.5

Rewrite the expression.

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

Step 7.1.5.4

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

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Step 7.1.5.4.1

Cancel the common factor.

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

Step 7.1.5.4.2

Rewrite the expression.

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

Step 7.1.6

Rewrite the equation.

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

Step 7.2

Integrate both sides.

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Step 7.2.1

Set up an integral on each side.

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

Step 7.2.2

Integrate the left side.

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Step 7.2.2.1

Let $u = 2 V^{2} + 1$ . Then $d u = 4 V d V$ , so $\frac{1}{4} d u = V d V$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.2.2.1.1

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

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Step 7.2.2.1.1.1

Differentiate $2 V^{2} + 1$ .

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

Step 7.2.2.1.1.2

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

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

Step 7.2.2.1.1.3

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

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Step 7.2.2.1.1.3.1

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

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

Step 7.2.2.1.1.3.2

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

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

Step 7.2.2.1.1.3.3

Multiply $2$ by $2$ .

$4 V + \frac{d}{d V} [1]$

Step 7.2.2.1.1.4

Differentiate using the Constant Rule.

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Step 7.2.2.1.1.4.1

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

$4 V + 0$

Step 7.2.2.1.1.4.2

Add $4 V$ and $0$ .

$4 V$

Step 7.2.2.1.2

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

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

Step 7.2.2.2

Simplify.

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Step 7.2.2.2.1

Multiply $\frac{1}{u}$ by $\frac{1}{4}$ .

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

Step 7.2.2.2.2

Move $4$ to the left of $u$ .

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

Step 7.2.2.3

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

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

Step 7.2.2.4

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 7.2.2.5

Simplify.

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

Step 7.2.2.6

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

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

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.

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

Step 7.2.3.2

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

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

Step 7.2.3.3

Simplify.

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

Step 7.2.4

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

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

Step 7.3

Solve for $V$ .

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Step 7.3.1

Multiply both sides of the equation by $4$ .

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

Step 7.3.2

Simplify both sides of the equation.

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Step 7.3.2.1

Simplify the left side.

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Step 7.3.2.1.1

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

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Step 7.3.2.1.1.1

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

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

Step 7.3.2.1.1.2

Cancel the common factor of $4$ .

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Step 7.3.2.1.1.2.1

Cancel the common factor.

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

Step 7.3.2.1.1.2.2

Rewrite the expression.

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

Step 7.3.2.2

Simplify the right side.

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Step 7.3.2.2.1

Simplify $4 (- \ln (| x |) + C)$ .

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Step 7.3.2.2.1.1

Apply the distributive property.

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

Step 7.3.2.2.1.2

Multiply $- 1$ by $4$ .

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

Step 7.3.3

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

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

Step 7.3.4

Simplify the left side.

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Step 7.3.4.1

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

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Step 7.3.4.1.1

Simplify each term.

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Step 7.3.4.1.1.1

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

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

Step 7.3.4.1.1.2

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

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

Step 7.3.4.1.2

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

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

Step 7.3.4.1.3

Reorder factors in $\ln (| 2 V^{2} + 1 | x^{4})$ .

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

Step 7.3.5

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

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

Step 7.3.6

Rewrite $\ln (x^{4} | 2 V^{2} + 1 |) = 4 C$ 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 $b^{y} = x$ .

$e^{4 C} = x^{4} | 2 V^{2} + 1 |$

Step 7.3.7

Solve for $V$ .

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Step 7.3.7.1

Rewrite the equation as $x^{4} | 2 V^{2} + 1 | = e^{4 C}$ .

$x^{4} | 2 V^{2} + 1 | = e^{4 C}$

Step 7.3.7.2

Divide each term in $x^{4} | 2 V^{2} + 1 | = e^{4 C}$ by $x^{4}$ and simplify.

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Step 7.3.7.2.1

Divide each term in $x^{4} | 2 V^{2} + 1 | = e^{4 C}$ by $x^{4}$ .

$\frac{x^{4} | 2 V^{2} + 1 |}{x^{4}} = \frac{e^{4 C}}{x^{4}}$

Step 7.3.7.2.2

Simplify the left side.

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Step 7.3.7.2.2.1

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

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Step 7.3.7.2.2.1.1

Cancel the common factor.

$\frac{x^{4} | 2 V^{2} + 1 |}{x^{4}} = \frac{e^{4 C}}{x^{4}}$

Step 7.3.7.2.2.1.2

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

$| 2 V^{2} + 1 | = \frac{e^{4 C}}{x^{4}}$

Step 7.3.7.3

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

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

Step 7.3.7.4

Subtract $1$ from both sides of the equation.

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

Step 7.3.7.5

Divide each term in $2 V^{2} = \pm \frac{e^{4 C}}{x^{4}} - 1$ by $2$ and simplify.

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Step 7.3.7.5.1

Divide each term in $2 V^{2} = \pm \frac{e^{4 C}}{x^{4}} - 1$ by $2$ .

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

Step 7.3.7.5.2

Simplify the left side.

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Step 7.3.7.5.2.1

Cancel the common factor of $2$ .

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Step 7.3.7.5.2.1.1

Cancel the common factor.

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

Step 7.3.7.5.2.1.2

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

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

Step 7.3.7.5.3

Simplify the right side.

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Step 7.3.7.5.3.1

Move the negative in front of the fraction.

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

Step 7.3.7.6

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

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

Step 7.3.7.7

Simplify $\pm \sqrt{\frac{\pm \frac{e^{4 C}}{x^{4}}}{2} - \frac{1}{2}}$ .

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Step 7.3.7.7.1

Combine the numerators over the common denominator.

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

Step 7.3.7.7.2

Rewrite $\sqrt{\frac{\pm \frac{e^{4 C}}{x^{4}} - 1}{2}}$ as $\frac{\sqrt{\pm \frac{e^{4 C}}{x^{4}} - 1}}{\sqrt{2}}$ .

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

Step 7.3.7.7.3

Multiply $\frac{\sqrt{\pm \frac{e^{4 C}}{x^{4}} - 1}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 7.3.7.7.4

Combine and simplify the denominator.

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Step 7.3.7.7.4.1

Multiply $\frac{\sqrt{\pm \frac{e^{4 C}}{x^{4}} - 1}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 7.3.7.7.4.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 7.3.7.7.4.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 7.3.7.7.4.4

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

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

Step 7.3.7.7.4.5

Add $1$ and $1$ .

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

Step 7.3.7.7.4.6

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

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Step 7.3.7.7.4.6.1

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

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

Step 7.3.7.7.4.6.2

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

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

Step 7.3.7.7.4.6.3

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

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

Step 7.3.7.7.4.6.4

Cancel the common factor of $2$ .

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Step 7.3.7.7.4.6.4.1

Cancel the common factor.

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

Step 7.3.7.7.4.6.4.2

Rewrite the expression.

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

Step 7.3.7.7.4.6.5

Evaluate the exponent.

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

Step 7.3.7.7.5

Combine using the product rule for radicals.

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

Step 7.3.7.7.6

Reorder factors in $\pm \frac{\sqrt{(\pm \frac{e^{4 C}}{x^{4}} - 1) \cdot 2}}{2}$ .

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

Step 7.4

Group the constant terms together.

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Step 7.4.1

Simplify the constant of integration.

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

Step 7.4.2

Combine constants with the plus or minus.

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

Step 8

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

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

Step 9

Solve $\frac{y}{x} = \pm \frac{\sqrt{2 (C \frac{1}{x^{4}} - 1)}}{2}$ for $y$ .

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Step 9.1

Multiply both sides by $x$ .

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

Step 9.2

Simplify.

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Step 9.2.1

Simplify the left side.

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Step 9.2.1.1

Cancel the common factor of $x$ .

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Step 9.2.1.1.1

Cancel the common factor.

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

Step 9.2.1.1.2

Rewrite the expression.

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

Step 9.2.2

Simplify the right side.

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Step 9.2.2.1

Simplify $(\pm \frac{\sqrt{2 (C (\frac{1}{x^{4}}) - 1)}}{2}) x$ .

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Step 9.2.2.1.1

Simplify the numerator.

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Step 9.2.2.1.1.1

Combine $C$ and $\frac{1}{x^{4}}$ .

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

Step 9.2.2.1.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x^{4}}{x^{4}}$ .

$y = (\pm \frac{\sqrt{2 (\frac{C}{x^{4}} - 1 \cdot \frac{x^{4}}{x^{4}})}}{2}) x$

Step 9.2.2.1.1.3

Combine $- 1$ and $\frac{x^{4}}{x^{4}}$ .

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

Step 9.2.2.1.1.4

Combine the numerators over the common denominator.

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

Step 9.2.2.1.1.5

Combine $2$ and $\frac{C - x^{4}}{x^{4}}$ .

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

Step 9.2.2.1.1.6

Rewrite $\frac{2 (C - x^{4})}{x^{4}}$ as ${(\frac{1}{x^{2}})}^{2} (2 (C - x^{4}))$ .

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Step 9.2.2.1.1.6.1

Factor the perfect power $1^{2}$ out of $2 (C - x^{4})$ .

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

Step 9.2.2.1.1.6.2

Factor the perfect power ${(x^{2})}^{2}$ out of $x^{4}$ .

$y = (\pm \frac{\sqrt{\frac{1^{2} (2 (C - x^{4}))}{{(x^{2})}^{2} \cdot 1}}}{2}) x$

Step 9.2.2.1.1.6.3

Rearrange the fraction $\frac{1^{2} (2 (C - x^{4}))}{{(x^{2})}^{2} \cdot 1}$ .

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

Step 9.2.2.1.1.7

Pull terms out from under the radical.

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

Step 9.2.2.1.1.8

Combine $\frac{1}{x^{2}}$ and $\sqrt{2 (C - x^{4})}$ .

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

Step 9.2.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$y = (\pm \frac{\sqrt{2 (C - x^{4})}}{x^{2}} \cdot \frac{1}{2}) x$

Step 9.2.2.1.3

Combine.

$y = (\pm \frac{\sqrt{2 (C - x^{4})} \cdot 1}{x^{2} \cdot 2}) x$

Step 9.2.2.1.4

Simplify the expression.

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Step 9.2.2.1.4.1

Multiply $\sqrt{2 (C - x^{4})}$ by $1$ .

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

Step 9.2.2.1.4.2

Move $2$ to the left of $x^{2}$ .

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