Calculus Examples

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

Step 1

Subtract $x (1 + y^{2}) d x$ from both sides of the equation.

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Combine $\frac{1}{1 + y^{2}}$ and $y$ .

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.4.4

Cancel the common factor.

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

Step 3.4.5

Rewrite the expression.

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

Step 3.5

Combine $\frac{- 1}{1 + x^{2}}$ and $x$ .

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

Step 3.6

Move the negative in front of the fraction.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

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 4.2.1.1

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

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Step 4.2.1.1.1

Differentiate $1 + y^{2}$ .

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

Step 4.2.1.1.2

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 4.2.1.1.3

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 4.2.1.1.4

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 4.2.1.1.5

Add $0$ and $2 y$ .

$2 y$

Step 4.2.1.2

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

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

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

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

Step 4.2.5

Simplify.

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

Step 4.2.6

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

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

Step 4.3

Integrate the right side.

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Step 4.3.1

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

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

Step 4.3.2

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 4.3.2.1

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

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Step 4.3.2.1.1

Differentiate $1 + x^{2}$ .

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

Step 4.3.2.1.2

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 4.3.2.1.3

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 4.3.2.1.4

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 4.3.2.1.5

Add $0$ and $2 x$ .

$2 x$

Step 4.3.2.2

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

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

Step 4.3.3

Simplify.

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Step 4.3.3.1

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

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

Step 4.3.3.2

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

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

Step 4.3.4

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

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

Step 4.3.5

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

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

Step 4.3.6

Simplify.

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

Step 4.3.7

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

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

Step 4.4

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

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

Step 5

Solve for $y$ .

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Step 5.1

Multiply both sides of the equation by $2$ .

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

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

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

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Step 5.2.1.1.1

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

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

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

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

Step 5.2.1.1.2.2

Rewrite the expression.

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

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

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Step 5.2.2.1.1

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

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

Step 5.2.2.1.2

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

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

Step 5.2.2.1.3

Simplify terms.

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Step 5.2.2.1.3.1

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

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

Step 5.2.2.1.3.2

Combine the numerators over the common denominator.

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

Step 5.2.2.1.3.3

Cancel the common factor of $2$ .

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Step 5.2.2.1.3.3.1

Cancel the common factor.

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

Step 5.2.2.1.3.3.2

Rewrite the expression.

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

Step 5.2.2.1.4

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

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

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Step 5.6.1

Apply the distributive property.

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

Step 5.6.2

Apply the distributive property.

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

Step 5.6.3

Apply the distributive property.

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

Step 5.7

Simplify each term.

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Step 5.7.1

Multiply $1$ by $1$ .

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

Step 5.7.2

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

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

Step 5.7.3

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

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

Step 5.8

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

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

Step 5.9

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

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

Step 5.10

Solve for $y$ .

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Step 5.10.1

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

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

Step 5.10.2

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

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

Step 5.10.3

Move all terms not containing $y$ to the right side of the equation.

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Step 5.10.3.1

Subtract $1$ from both sides of the equation.

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

Step 5.10.3.2

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

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

Step 5.10.4

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

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Step 5.10.4.1

Multiply by $1$ .

$y^{2} \cdot 1 + y^{2} x^{2} = \pm e^{2 C} - 1 - x^{2}$

Step 5.10.4.2

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

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

Step 5.10.4.3

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

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

Step 5.10.5

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

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Step 5.10.5.1

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

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

Step 5.10.5.2

Simplify the left side.

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Step 5.10.5.2.1

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

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Step 5.10.5.2.1.1

Cancel the common factor.

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

Step 5.10.5.2.1.2

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

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

Step 5.10.5.3

Simplify the right side.

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Step 5.10.5.3.1

Simplify each term.

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Step 5.10.5.3.1.1

Move the negative in front of the fraction.

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

Step 5.10.5.3.1.2

Move the negative in front of the fraction.

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

Step 5.10.5.3.2

Combine into one fraction.

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Step 5.10.5.3.2.1

Combine the numerators over the common denominator.

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

Step 5.10.5.3.2.2

Combine the numerators over the common denominator.

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

Step 5.10.6

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

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

Step 5.10.7

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

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Step 5.10.7.1

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

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

Step 5.10.7.2

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

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

Step 5.10.7.3

Combine and simplify the denominator.

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Step 5.10.7.3.1

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

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

Step 5.10.7.3.2

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

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

Step 5.10.7.3.3

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

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

Step 5.10.7.3.4

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

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

Step 5.10.7.3.5

Add $1$ and $1$ .

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

Step 5.10.7.3.6

Rewrite ${\sqrt{1 + x^{2}}}^{2}$ as $1 + x^{2}$ .

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Step 5.10.7.3.6.1

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

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

Step 5.10.7.3.6.2

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

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

Step 5.10.7.3.6.3

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

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

Step 5.10.7.3.6.4

Cancel the common factor of $2$ .

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Step 5.10.7.3.6.4.1

Cancel the common factor.

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

Step 5.10.7.3.6.4.2

Rewrite the expression.

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

Step 5.10.7.3.6.5

Simplify.

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

Step 5.10.7.4

Combine using the product rule for radicals.

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

Step 6

Simplify the constant of integration.

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