Calculus Examples

$\frac{d x}{d y} + \frac{x}{y} = \frac{3}{x y}$

Step 1

Separate the variables.

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Step 1.1

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

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Step 1.1.1

Subtract $\frac{3}{x y}$ from both sides of the equation.

$\frac{d x}{d y} + \frac{x}{y} - \frac{3}{x y} = 0$

Step 1.1.2

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

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Step 1.1.2.1

Subtract $\frac{x}{y}$ from both sides of the equation.

$\frac{d x}{d y} - \frac{3}{x y} = - \frac{x}{y}$

Step 1.1.2.2

Add $\frac{3}{x y}$ to both sides of the equation.

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

Step 1.2

Factor.

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Step 1.2.1

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

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

Step 1.2.2

Write each expression with a common denominator of $y x$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.2.1

Multiply $\frac{x}{y}$ by $\frac{x}{x}$ .

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

Step 1.2.2.2

Reorder the factors of $y x$ .

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

Step 1.2.3

Combine the numerators over the common denominator.

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

Step 1.2.4

Multiply $x$ by $x$ by adding the exponents.

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Step 1.2.4.1

Move $x$ .

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

Step 1.2.4.2

Multiply $x$ by $x$ .

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

Step 1.3

Regroup factors.

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

Step 1.4

Multiply both sides by $\frac{x}{- x^{2} + 3}$ .

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

Step 1.5

Simplify.

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Step 1.5.1

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

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

Step 1.5.2

Cancel the common factor of $x$ .

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Step 1.5.2.1

Factor $x$ out of $x y$ .

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

Step 1.5.2.2

Cancel the common factor.

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

Step 1.5.2.3

Rewrite the expression.

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

Step 1.5.3

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

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

Step 1.5.4

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

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Step 1.5.4.1

Cancel the common factor.

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

Step 1.5.4.2

Rewrite the expression.

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

Step 1.6

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u = - x^{2} + 3$ . Then $d u = - 2 x d x$ , so $- \frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1.1

Let $u = - x^{2} + 3$ . Find $\frac{d u}{d x}$ .

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Step 2.2.1.1.1

Differentiate $- x^{2} + 3$ .

$\frac{d}{d x} [- x^{2} + 3]$

Step 2.2.1.1.2

By the Sum Rule, the derivative of $- x^{2} + 3$ with respect to $x$ is $\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [3]$

Step 2.2.1.1.3

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

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Step 2.2.1.1.3.1

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

$- \frac{d}{d x} [x^{2}] + \frac{d}{d x} [3]$

Step 2.2.1.1.3.2

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

$- (2 x) + \frac{d}{d x} [3]$

Step 2.2.1.1.3.3

Multiply $2$ by $- 1$ .

$- 2 x + \frac{d}{d x} [3]$

Step 2.2.1.1.4

Differentiate using the Constant Rule.

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Step 2.2.1.1.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$- 2 x + 0$

Step 2.2.1.1.4.2

Add $- 2 x$ and $0$ .

$- 2 x$

Step 2.2.1.2

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

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

Step 2.2.2

Simplify.

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Step 2.2.2.1

Move the negative in front of the fraction.

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

Simplify.

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

Step 2.2.7

Replace all occurrences of $u$ with $- x^{2} + 3$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Solve for $x$ .

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Step 3.1

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

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

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

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Step 3.2.1.1.1

Combine $\ln (| - x^{2} + 3 |)$ and $\frac{1}{2}$ .

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

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 $- \frac{\ln (| - x^{2} + 3 |)}{2}$ into the numerator.

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

Step 3.2.1.1.2.2

Factor $2$ out of $- 2$ .

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

Step 3.2.1.1.2.3

Cancel the common factor.

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

Step 3.2.1.1.2.4

Rewrite the expression.

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

Step 3.2.1.1.3

Multiply.

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Step 3.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2.1.1.3.2

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

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Apply the distributive property.

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

Step 3.3

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

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

Step 3.4

Simplify the left side.

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Step 3.4.1

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

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Step 3.4.1.1

Simplify each term.

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Step 3.4.1.1.1

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

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

Step 3.4.1.1.2

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

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

Step 3.4.1.2

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

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

Step 3.4.1.3

Reorder factors in $\ln (| - x^{2} + 3 | y^{2})$ .

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Solve for $x$ .

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Step 3.7.1

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

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

Step 3.7.2

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

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Step 3.7.2.1

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

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

Step 3.7.2.2

Simplify the left side.

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Step 3.7.2.2.1

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

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Step 3.7.2.2.1.1

Cancel the common factor.

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

Step 3.7.2.2.1.2

Divide $| - x^{2} + 3 |$ by $1$ .

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

Step 3.7.3

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

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

Step 3.7.4

Subtract $3$ from both sides of the equation.

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

Step 3.7.5

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

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Step 3.7.5.1

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

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

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.

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

Step 3.7.5.2.2

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

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

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 $\frac{\pm \frac{e^{- 2 C}}{y^{2}}}{- 1}$ .

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

Step 3.7.5.3.1.2

Rewrite $- 1 \cdot (\pm \frac{e^{- 2 C}}{y^{2}})$ as $- (\pm \frac{e^{- 2 C}}{y^{2}})$ .

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

Step 3.7.5.3.1.3

Divide $- 3$ by $- 1$ .

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

Step 3.7.6

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

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

Step 4

Group the constant terms together.

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Step 4.1

Simplify the constant of integration.

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

Step 4.2

Combine constants with the plus or minus.

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