Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Rewrite using the commutative property of multiplication.

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

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

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

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

Step 3.2.2

Factor $x$ out of $x y$ .

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 3.3

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

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

Step 3.4

Move the negative in front of the fraction.

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

Step 3.5

Rewrite using the commutative property of multiplication.

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

Step 3.6

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

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Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

Cancel the common factor.

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

Step 3.6.4

Rewrite the expression.

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

Step 3.7

Move the negative in front of the fraction.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

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

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

Step 4.2.2

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

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Step 4.2.2.1

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

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Step 4.2.2.1.1

Differentiate $y^{2} + 2$ .

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

Step 4.2.2.1.2

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

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

Step 4.2.2.1.3

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

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

Step 4.2.2.1.4

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

$2 y + 0$

Step 4.2.2.1.5

Add $2 y$ and $0$ .

$2 y$

Step 4.2.2.2

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

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

Step 4.2.3

Simplify.

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Step 4.2.3.1

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

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

Step 4.2.3.2

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

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

Step 4.2.4

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

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

Step 4.2.5

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

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

Step 4.2.6

Simplify.

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

Step 4.2.7

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

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

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.

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

Step 4.3.2

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

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

Step 4.3.3

Simplify.

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

Step 4.4

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

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

Step 5

Solve for $y$ .

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Step 5.1

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

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

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

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Step 5.2.1.1.1

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

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

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

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

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

Step 5.2.1.1.2.2

Factor $2$ out of $- 2$ .

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

Step 5.2.1.1.2.3

Cancel the common factor.

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

Step 5.2.1.1.2.4

Rewrite the expression.

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

Step 5.2.1.1.3

Multiply.

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Step 5.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2.1.1.3.2

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

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $- 2 (- \ln (| x |) + C)$ .

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Step 5.2.2.1.1

Apply the distributive property.

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

Step 5.2.2.1.2

Multiply $- 1$ by $- 2$ .

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

Step 5.3

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

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

Step 5.4

Simplify the left side.

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Step 5.4.1

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

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Step 5.4.1.1

Simplify each term.

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Step 5.4.1.1.1

Simplify $- 2 \ln (| x |)$ by moving $2$ inside the logarithm.

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

Step 5.4.1.1.2

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

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

Step 5.4.1.2

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

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

Solve for $y$ .

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Step 5.7.1

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

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

Step 5.7.2

Multiply both sides by $x^{2}$ .

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

Step 5.7.3

Simplify.

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Step 5.7.3.1

Simplify the left side.

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Step 5.7.3.1.1

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

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Step 5.7.3.1.1.1

Cancel the common factor.

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

Step 5.7.3.1.1.2

Rewrite the expression.

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

Step 5.7.3.2

Simplify the right side.

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Step 5.7.3.2.1

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

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

Step 5.7.4

Solve for $y$ .

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Step 5.7.4.1

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

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

Step 5.7.4.2

Subtract $2$ from both sides of the equation.

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

Step 5.7.4.3

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

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

Step 6

Group the constant terms together.

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Step 6.1

Simplify the constant of integration.

$y = \pm \sqrt{\pm x^{2} C - 2}$

Step 6.2

Combine constants with the plus or minus.

$y = \pm \sqrt{C x^{2} - 2}$