Calculus Examples

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

Step 1

Separate the variables.

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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.

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

Step 1.1.1.2

Multiply $2$ by $1$ .

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

Step 1.1.1.3

Apply the distributive property.

$2 x + 2 x y^{2} - y \frac{d y}{d x} \cdot 1 - y \frac{d y}{d x} x^{2} = 0$

Step 1.1.1.4

Multiply $- 1$ by $1$ .

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Factor $y \frac{d y}{d x}$ out of $y \frac{d y}{d x} \cdot - 1 + y \frac{d y}{d x} (- 1 x^{2})$ .

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

Step 1.1.4

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

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

Step 1.1.5

Divide each term in $y \frac{d y}{d x} (- 1 - x^{2}) = - 2 x - 2 x y^{2}$ by $y (- 1 - x^{2})$ and simplify.

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Step 1.1.5.1

Divide each term in $y \frac{d y}{d x} (- 1 - x^{2}) = - 2 x - 2 x y^{2}$ by $y (- 1 - x^{2})$ .

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

Step 1.1.5.2

Simplify the left side.

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Step 1.1.5.2.1

Cancel the common factor of $y$ .

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Step 1.1.5.2.1.1

Cancel the common factor.

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

Step 1.1.5.2.1.2

Rewrite the expression.

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

Step 1.1.5.2.2

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

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Step 1.1.5.2.2.1

Cancel the common factor.

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

Step 1.1.5.2.2.2

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

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

Step 1.1.5.3

Simplify the right side.

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Step 1.1.5.3.1

Simplify each term.

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Step 1.1.5.3.1.1

Move the negative in front of the fraction.

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

Step 1.1.5.3.1.2

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

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Step 1.1.5.3.1.2.1

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

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

Step 1.1.5.3.1.2.2

Cancel the common factors.

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Step 1.1.5.3.1.2.2.1

Cancel the common factor.

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

Step 1.1.5.3.1.2.2.2

Rewrite the expression.

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

Step 1.1.5.3.1.3

Move the negative in front of the fraction.

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

Step 1.1.5.3.2

To write $- \frac{2 x y}{- 1 - x^{2}}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 1.1.5.3.3

Write each expression with a common denominator of $y (- 1 - x^{2})$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.5.3.3.1

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

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

Step 1.1.5.3.3.2

Reorder the factors of $(- 1 - x^{2}) y$ .

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

Step 1.1.5.3.4

Combine the numerators over the common denominator.

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

Step 1.1.5.3.5

Simplify the numerator.

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Step 1.1.5.3.5.1

Factor $2 x$ out of $- 2 x - 2 x y \cdot y$ .

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Step 1.1.5.3.5.1.1

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

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

Step 1.1.5.3.5.1.2

Factor $2 x$ out of $- 2 x y \cdot y$ .

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

Step 1.1.5.3.5.1.3

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

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

Step 1.1.5.3.5.2

Combine exponents.

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Step 1.1.5.3.5.2.1

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

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

Step 1.1.5.3.5.2.2

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

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

Step 1.1.5.3.5.2.3

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

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

Step 1.1.5.3.5.2.4

Add $1$ and $1$ .

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

Step 1.1.5.3.5.3

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

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

Step 1.1.5.3.6

Simplify terms.

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Step 1.1.5.3.6.1

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

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

Step 1.1.5.3.6.2

Factor $- 1$ out of $- y^{2}$ .

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

Step 1.1.5.3.6.3

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

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

Step 1.1.5.3.6.4

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

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

Step 1.1.5.3.6.5

Factor $- 1$ out of $- x^{2}$ .

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

Step 1.1.5.3.6.6

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

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

Step 1.1.5.3.6.7

Cancel the common factor.

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

Step 1.1.5.3.6.8

Rewrite the expression.

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

Step 1.2

Regroup factors.

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

Step 1.3

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

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

Step 1.4

Simplify.

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Step 1.4.1

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

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

Step 1.4.2

Cancel the common factor of $y$ .

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Step 1.4.2.1

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

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

Step 1.4.2.2

Cancel the common factor.

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

Step 1.4.2.3

Rewrite the expression.

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

Step 1.4.3

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

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Step 1.4.3.1

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

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

Step 1.4.3.2

Cancel the common factor.

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

Step 1.4.3.3

Rewrite the expression.

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

Step 1.5

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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Step 2.2.1.1.1

Differentiate $1 + y^{2}$ .

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

Step 2.2.1.1.2

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

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

Step 2.2.1.1.3

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

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

Step 2.2.1.1.4

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

$0 + 2 y$

Step 2.2.1.1.5

Add $0$ and $2 y$ .

$2 y$

Step 2.2.1.2

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

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

Step 2.2.2

Simplify.

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Simplify.

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

Step 2.2.6

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

Let $u_{2} = 1 + x^{2}$ . 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 $u_{2} = 1 + x^{2}$ . Find $\frac{d u_{2}}{d x}$ .

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Step 2.3.2.1.1

Differentiate $1 + x^{2}$ .

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

Step 2.3.2.1.2

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

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

Step 2.3.2.1.3

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

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

Step 2.3.2.1.4

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

$0 + 2 x$

Step 2.3.2.1.5

Add $0$ and $2 x$ .

$2 x$

Step 2.3.2.2

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

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

Step 2.3.3

Simplify.

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Step 2.3.3.1

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

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

Step 2.3.3.2

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

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

Step 2.3.4

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

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

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

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

Step 2.3.5.2

Cancel the common factor of $2$ .

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Step 2.3.5.2.1

Cancel the common factor.

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

Step 2.3.5.2.2

Rewrite the expression.

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

Step 2.3.5.3

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

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

Step 2.3.6

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

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

Step 2.3.7

Replace all occurrences of $u_{2}$ with $1 + x^{2}$ .

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

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

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

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Apply the distributive property.

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

Step 3.3

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

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

Step 3.4

Simplify the left side.

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Step 3.4.1

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

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Step 3.4.1.1

Simplify each term.

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Step 3.4.1.1.1

Simplify $- 2 \ln (| 1 + x^{2} |)$ by moving $2$ inside the logarithm.

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

Step 3.4.1.1.2

Remove the absolute value in ${| 1 + x^{2} |}^{2}$ because exponentiations with even powers are always positive.

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

Step 3.4.1.2

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Solve for $y$ .

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Step 3.7.1

Rewrite the equation as $\frac{| 1 + y^{2} |}{{(1 + x^{2})}^{2}} = e^{2 C}$ .

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

Step 3.7.2

Multiply both sides by ${(1 + x^{2})}^{2}$ .

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

Step 3.7.3

Simplify the left side.

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Step 3.7.3.1

Cancel the common factor of ${(1 + x^{2})}^{2}$ .

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Step 3.7.3.1.1

Cancel the common factor.

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

Step 3.7.3.1.2

Rewrite the expression.

$| 1 + y^{2} | = e^{2 C} {(1 + x^{2})}^{2}$

Step 3.7.4

Solve for $y$ .

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Step 3.7.4.1

Simplify $e^{2 C} {(1 + x^{2})}^{2}$ .

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Step 3.7.4.1.1

Rewrite ${(1 + x^{2})}^{2}$ as $(1 + x^{2}) (1 + x^{2})$ .

$| 1 + y^{2} | = e^{2 C} ((1 + x^{2}) (1 + x^{2}))$

Step 3.7.4.1.2

Expand $(1 + x^{2}) (1 + x^{2})$ using the FOIL Method.

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Step 3.7.4.1.2.1

Apply the distributive property.

$| 1 + y^{2} | = e^{2 C} (1 (1 + x^{2}) + x^{2} (1 + x^{2}))$

Step 3.7.4.1.2.2

Apply the distributive property.

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

Step 3.7.4.1.2.3

Apply the distributive property.

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

Step 3.7.4.1.3

Simplify and combine like terms.

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Step 3.7.4.1.3.1

Simplify each term.

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Step 3.7.4.1.3.1.1

Multiply $1$ by $1$ .

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

Step 3.7.4.1.3.1.2

Multiply $x^{2}$ by $1$ .

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

Step 3.7.4.1.3.1.3

Multiply $x^{2}$ by $1$ .

$| 1 + y^{2} | = e^{2 C} (1 + x^{2} + x^{2} + x^{2} x^{2})$

Step 3.7.4.1.3.1.4

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 3.7.4.1.3.1.4.1

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

$| 1 + y^{2} | = e^{2 C} (1 + x^{2} + x^{2} + x^{2 + 2})$

Step 3.7.4.1.3.1.4.2

Add $2$ and $2$ .

$| 1 + y^{2} | = e^{2 C} (1 + x^{2} + x^{2} + x^{4})$

Step 3.7.4.1.3.2

Add $x^{2}$ and $x^{2}$ .

$| 1 + y^{2} | = e^{2 C} (1 + 2 x^{2} + x^{4})$

Step 3.7.4.1.4

Apply the distributive property.

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

Step 3.7.4.1.5

Simplify.

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Step 3.7.4.1.5.1

Multiply $e^{2 C}$ by $1$ .

$| 1 + y^{2} | = e^{2 C} + e^{2 C} (2 x^{2}) + e^{2 C} x^{4}$

Step 3.7.4.1.5.2

Rewrite using the commutative property of multiplication.

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

Step 3.7.4.1.6

Reorder factors in $e^{2 C} + 2 e^{2 C} x^{2} + e^{2 C} x^{4}$ .

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

Step 3.7.4.1.7

Move $e^{2 C}$ .

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

Step 3.7.4.1.8

Reorder $2 x^{2} e^{2 C}$ and $x^{4} e^{2 C}$ .

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

Step 3.7.4.2

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

$1 + y^{2} = \pm (x^{4} e^{2 C} + 2 x^{2} e^{2 C} + e^{2 C})$

Step 3.7.4.3

Subtract $1$ from both sides of the equation.

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

Step 3.7.4.4

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

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

Step 4

Simplify the constant of integration.

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