Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Factor.

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Step 1.1.1

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

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Step 1.1.1.1

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

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

Step 1.1.1.2

Factor $3 x$ out of $- 3 x$ .

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

Step 1.1.1.3

Factor $3 x$ out of $3 x (- 1 y^{2}) + 3 x (- 1)$ .

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Factor $y$ out of $2 y$ .

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

Step 1.2.3

Factor $y$ out of $y x^{2} + y \cdot 2$ .

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

Step 1.3

Regroup factors.

$\frac{d y}{d x} = \frac{3 x}{x^{2} + 2} \cdot \frac{- y^{2} - 1}{y}$

Step 1.4

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

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

Step 1.5

Simplify.

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Step 1.5.1

Multiply $\frac{3 x}{x^{2} + 2}$ by $\frac{- y^{2} - 1}{y}$ .

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

Step 1.5.2

Cancel the common factor of $y$ .

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Step 1.5.2.1

Factor $y$ out of $(x^{2} + 2) y$ .

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

Step 1.5.2.2

Cancel the common factor.

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

Step 1.5.2.3

Rewrite the expression.

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

Step 1.5.3

Cancel the common factor of $- y^{2} - 1$ .

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Step 1.5.3.1

Factor $- y^{2} - 1$ out of $3 x (- y^{2} - 1)$ .

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

Step 1.5.3.2

Cancel the common factor.

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

Step 1.5.3.3

Rewrite the expression.

$\frac{y}{- y^{2} - 1} \frac{d y}{d x} = \frac{3 x}{x^{2} + 2}$

Step 1.6

Rewrite the equation.

$\frac{y}{- y^{2} - 1} d y = \frac{3 x}{x^{2} + 2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u_{1} = - y^{2} - 1$ . Then $d u_{1} = - 2 y d y$ , so $- \frac{1}{2} d u_{1} = y d y$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.2.1.1

Let $u_{1} = - y^{2} - 1$ . Find $\frac{d u_{1}}{d y}$ .

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Step 2.2.1.1.1

Differentiate $- y^{2} - 1$ .

$\frac{d}{d y} [- y^{2} - 1]$

Step 2.2.1.1.2

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

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

Step 2.2.1.1.3

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

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Step 2.2.1.1.3.1

Since $- 1$ is constant with respect to $y$ , the derivative of $- y^{2}$ with respect to $y$ is $- \frac{d}{d y} [y^{2}]$ .

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

Step 2.2.1.1.3.2

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

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

Step 2.2.1.1.3.3

Multiply $2$ by $- 1$ .

$- 2 y + \frac{d}{d y} [- 1]$

Step 2.2.1.1.4

Differentiate using the Constant Rule.

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Step 2.2.1.1.4.1

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

$- 2 y + 0$

Step 2.2.1.1.4.2

Add $- 2 y$ and $0$ .

$- 2 y$

Step 2.2.1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \frac{1}{u_{1}} \cdot \frac{1}{- 2} d u_{1} = \int \frac{3 x}{x^{2} + 2} d x$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Move the negative in front of the fraction.

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

Step 2.2.2.2

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

$\int - \frac{1}{u_{1} \cdot 2} d u_{1} = \int \frac{3 x}{x^{2} + 2} d x$

Step 2.2.2.3

Move $2$ to the left of $u_{1}$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

Simplify.

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

Step 2.2.7

Replace all occurrences of $u_{1}$ with $- y^{2} - 1$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 2.3.2

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

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Step 2.3.2.1

Let $u_{2} = x^{2} + 2$ . Find $\frac{d u_{2}}{d x}$ .

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Step 2.3.2.1.1

Differentiate $x^{2} + 2$ .

$\frac{d}{d x} [x^{2} + 2]$

Step 2.3.2.1.2

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

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2]$

Step 2.3.2.1.3

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

$2 x + \frac{d}{d x} [2]$

Step 2.3.2.1.4

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

$2 x + 0$

Step 2.3.2.1.5

Add $2 x$ and $0$ .

$2 x$

Step 2.3.2.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

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

Step 2.3.3

Simplify.

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Step 2.3.3.1

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

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

Step 2.3.3.2

Move $2$ to the left of $u_{2}$ .

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

Step 2.3.4

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

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

Step 2.3.5

Combine $\frac{1}{2}$ and $3$ .

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

Step 2.3.6

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

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

Step 2.3.7

Simplify.

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

Step 2.3.8

Replace all occurrences of $u_{2}$ with $x^{2} + 2$ .

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $- 2$ .

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

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Move the leading negative in $- \frac{\ln (| - y^{2} - 1 |)}{2}$ into the numerator.

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

Step 3.2.1.1.2.2

Factor $2$ out of $- 2$ .

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

Step 3.2.1.1.2.3

Cancel the common factor.

$2 \cdot - 1 \frac{- \ln (| - y^{2} - 1 |)}{2} = - 2 (\frac{3}{2} \ln (| x^{2} + 2 |) + C)$

Step 3.2.1.1.2.4

Rewrite the expression.

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

Step 3.2.1.1.3

Multiply.

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Step 3.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2.1.1.3.2

Multiply $\ln (| - y^{2} - 1 |)$ by $1$ .

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

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

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

Step 3.2.2.1.2

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

$\ln (| - y^{2} - 1 |) = - 2 (\frac{3 \ln (| x^{2} + 2 |)}{2} + C \cdot \frac{2}{2})$

Step 3.2.2.1.3

Simplify terms.

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Step 3.2.2.1.3.1

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

$\ln (| - y^{2} - 1 |) = - 2 (\frac{3 \ln (| x^{2} + 2 |)}{2} + \frac{C \cdot 2}{2})$

Step 3.2.2.1.3.2

Combine the numerators over the common denominator.

$\ln (| - y^{2} - 1 |) = - 2 \frac{3 \ln (| x^{2} + 2 |) + C \cdot 2}{2}$

Step 3.2.2.1.3.3

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.3.1

Factor $2$ out of $- 2$ .

$\ln (| - y^{2} - 1 |) = 2 (- 1) \frac{3 \ln (| x^{2} + 2 |) + C \cdot 2}{2}$

Step 3.2.2.1.3.3.2

Cancel the common factor.

$\ln (| - y^{2} - 1 |) = 2 \cdot - 1 \frac{3 \ln (| x^{2} + 2 |) + C \cdot 2}{2}$

Step 3.2.2.1.3.3.3

Rewrite the expression.

$\ln (| - y^{2} - 1 |) = - (3 \ln (| x^{2} + 2 |) + C \cdot 2)$

Step 3.2.2.1.4

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

$\ln (| - y^{2} - 1 |) = - (3 \ln (| x^{2} + 2 |) + 2 C)$

Step 3.2.2.1.5

Simplify by multiplying through.

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Step 3.2.2.1.5.1

Apply the distributive property.

$\ln (| - y^{2} - 1 |) = - (3 \ln (| x^{2} + 2 |)) - (2 C)$

Step 3.2.2.1.5.2

Multiply.

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Step 3.2.2.1.5.2.1

Multiply $3$ by $- 1$ .

$\ln (| - y^{2} - 1 |) = - 3 \ln (| x^{2} + 2 |) - (2 C)$

Step 3.2.2.1.5.2.2

Multiply $2$ by $- 1$ .

$\ln (| - y^{2} - 1 |) = - 3 \ln (| x^{2} + 2 |) - 2 C$

Step 3.3

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

$\ln (| - y^{2} - 1 |) + 3 \ln (| x^{2} + 2 |) = - 2 C$

Step 3.4

Simplify the left side.

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Step 3.4.1

Simplify $\ln (| - y^{2} - 1 |) + 3 \ln (| x^{2} + 2 |)$ .

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Step 3.4.1.1

Simplify $3 \ln (| x^{2} + 2 |)$ by moving $3$ inside the logarithm.

$\ln (| - y^{2} - 1 |) + \ln ({| x^{2} + 2 |}^{3}) = - 2 C$

Step 3.4.1.2

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

$\ln (| - y^{2} - 1 | {| x^{2} + 2 |}^{3}) = - 2 C$

Step 3.5

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

$e^{\ln (| - y^{2} - 1 | {| x^{2} + 2 |}^{3})} = e^{- 2 C}$

Step 3.6

Rewrite $\ln (| - y^{2} - 1 | {| x^{2} + 2 |}^{3}) = - 2 C$ 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 $b^{y} = x$ .

$e^{- 2 C} = | - y^{2} - 1 | {| x^{2} + 2 |}^{3}$

Step 3.7

Solve for $y$ .

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Step 3.7.1

Rewrite the equation as $| - y^{2} - 1 | {| x^{2} + 2 |}^{3} = e^{- 2 C}$ .

$| - y^{2} - 1 | {| x^{2} + 2 |}^{3} = e^{- 2 C}$

Step 3.7.2

Divide each term in $| - y^{2} - 1 | {| x^{2} + 2 |}^{3} = e^{- 2 C}$ by ${| x^{2} + 2 |}^{3}$ and simplify.

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Step 3.7.2.1

Divide each term in $| - y^{2} - 1 | {| x^{2} + 2 |}^{3} = e^{- 2 C}$ by ${| x^{2} + 2 |}^{3}$ .

$\frac{| - y^{2} - 1 | {| x^{2} + 2 |}^{3}}{{| x^{2} + 2 |}^{3}} = \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}}$

Step 3.7.2.2

Simplify the left side.

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Step 3.7.2.2.1

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

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Step 3.7.2.2.1.1

Cancel the common factor.

$\frac{| - y^{2} - 1 | {| x^{2} + 2 |}^{3}}{{| x^{2} + 2 |}^{3}} = \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}}$

Step 3.7.2.2.1.2

Divide $| - y^{2} - 1 |$ by $1$ .

$| - y^{2} - 1 | = \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}}$

Step 3.7.3

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

$- y^{2} - 1 = \pm \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}}$

Step 3.7.4

Add $1$ to both sides of the equation.

$- y^{2} = \pm \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}} + 1$

Step 3.7.5

Divide each term in $- y^{2} = \pm \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}} + 1$ by $- 1$ and simplify.

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Step 3.7.5.1

Divide each term in $- y^{2} = \pm \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}} + 1$ by $- 1$ .

$\frac{- y^{2}}{- 1} = \frac{\pm \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}}}{- 1} + \frac{1}{- 1}$

Step 3.7.5.2

Simplify the left side.

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Step 3.7.5.2.1

Dividing two negative values results in a positive value.

$\frac{y^{2}}{1} = \frac{\pm \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}}}{- 1} + \frac{1}{- 1}$

Step 3.7.5.2.2

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

$y^{2} = \frac{\pm \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}}}{- 1} + \frac{1}{- 1}$

Step 3.7.5.3

Simplify the right side.

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Step 3.7.5.3.1

Simplify each term.

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Step 3.7.5.3.1.1

Move the negative one from the denominator of $\frac{\pm \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}}}{- 1}$ .

$y^{2} = - 1 \cdot (\pm \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}}) + \frac{1}{- 1}$

Step 3.7.5.3.1.2

Rewrite $- 1 \cdot (\pm \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}})$ as $- (\pm \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}})$ .

$y^{2} = - (\pm \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}}) + \frac{1}{- 1}$

Step 3.7.5.3.1.3

Divide $1$ by $- 1$ .

$y^{2} = - (\pm \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}}) - 1$

Step 3.7.6

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

$y = \pm \sqrt{- (\pm \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}}) - 1}$

Step 3.7.7

Factor $- 1$ out of $- (\pm \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}}) - 1$ .

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Step 3.7.7.1

Rewrite $- 1$ as $- 1 (1)$ .

$y = \pm \sqrt{- (\pm \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}}) - 1 (1)}$

Step 3.7.7.2

Factor $- 1$ out of $- (\pm \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}}) - 1 (1)$ .

$y = \pm \sqrt{- (\pm \frac{e^{- 2 C}}{{| x^{2} + 2 |}^{3}} + 1)}$

Step 4

Group the constant terms together.

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Step 4.1

Simplify the constant of integration.

$y = \pm \sqrt{- (\pm \frac{C}{{| x^{2} + 2 |}^{3}} + 1)}$

Step 4.2

Combine constants with the plus or minus.

$y = \pm \sqrt{- (C \frac{1}{{| x^{2} + 2 |}^{3}} + 1)}$