Calculus Examples

$x^{3} \frac{d y}{d x} = \frac{6}{y}$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $x^{3} \frac{d y}{d x} = \frac{6}{y}$ by $x^{3}$ and simplify.

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Step 1.1.1

Divide each term in $x^{3} \frac{d y}{d x} = \frac{6}{y}$ by $x^{3}$ .

$\frac{x^{3} \frac{d y}{d x}}{x^{3}} = \frac{\frac{6}{y}}{x^{3}}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

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

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{x^{3} \frac{d y}{d x}}{x^{3}} = \frac{\frac{6}{y}}{x^{3}}$

Step 1.1.2.1.2

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

$\frac{d y}{d x} = \frac{\frac{6}{y}}{x^{3}}$

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{d y}{d x} = \frac{6}{y} \cdot \frac{1}{x^{3}}$

Step 1.1.3.2

Combine.

$\frac{d y}{d x} = \frac{6 \cdot 1}{y x^{3}}$

Step 1.1.3.3

Multiply $6$ by $1$ .

$\frac{d y}{d x} = \frac{6}{y x^{3}}$

Step 1.2

Regroup factors.

$\frac{d y}{d x} = \frac{6}{y} \cdot \frac{1}{x^{3}}$

Step 1.3

Multiply both sides by $\frac{y}{6}$ .

$\frac{y}{6} \frac{d y}{d x} = \frac{y}{6} (\frac{6}{y} \cdot \frac{1}{x^{3}})$

Step 1.4

Simplify.

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Step 1.4.1

Combine.

$\frac{y}{6} \frac{d y}{d x} = \frac{y}{6} \cdot \frac{6 \cdot 1}{y x^{3}}$

Step 1.4.2

Combine.

$\frac{y}{6} \frac{d y}{d x} = \frac{y (6 \cdot 1)}{6 (y x^{3})}$

Step 1.4.3

Cancel the common factor of $y$ .

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Step 1.4.3.1

Cancel the common factor.

$\frac{y}{6} \frac{d y}{d x} = \frac{y (6 \cdot 1)}{6 (y x^{3})}$

Step 1.4.3.2

Rewrite the expression.

$\frac{y}{6} \frac{d y}{d x} = \frac{6 \cdot 1}{6 (x^{3})}$

Step 1.4.4

Cancel the common factor of $6$ .

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Step 1.4.4.1

Cancel the common factor.

$\frac{y}{6} \frac{d y}{d x} = \frac{6 \cdot 1}{6 x^{3}}$

Step 1.4.4.2

Rewrite the expression.

$\frac{y}{6} \frac{d y}{d x} = \frac{1}{x^{3}}$

Step 1.5

Rewrite the equation.

$\frac{y}{6} d y = \frac{1}{x^{3}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{y}{6} d y = \int \frac{1}{x^{3}} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Since $\frac{1}{6}$ is constant with respect to $y$ , move $\frac{1}{6}$ out of the integral.

$\frac{1}{6} \int y d y = \int \frac{1}{x^{3}} d x$

Step 2.2.2

By the Power Rule, the integral of $y$ with respect to $y$ is $\frac{1}{2} y^{2}$ .

$\frac{1}{6} (\frac{1}{2} y^{2} + C_{1}) = \int \frac{1}{x^{3}} d x$

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

Rewrite $\frac{1}{6} (\frac{1}{2} y^{2} + C_{1})$ as $\frac{1}{6} \cdot \frac{1}{2} y^{2} + C_{1}$ .

$\frac{1}{6} \cdot \frac{1}{2} y^{2} + C_{1} = \int \frac{1}{x^{3}} d x$

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

Multiply $\frac{1}{6}$ by $\frac{1}{2}$ .

$\frac{1}{6 \cdot 2} y^{2} + C_{1} = \int \frac{1}{x^{3}} d x$

Step 2.2.3.2.2

Multiply $6$ by $2$ .

$\frac{1}{12} y^{2} + C_{1} = \int \frac{1}{x^{3}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Apply basic rules of exponents.

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Step 2.3.1.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{12} y^{2} + C_{1} = \int {(x^{3})}^{- 1} d x$

Step 2.3.1.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 2.3.1.2.1

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

$\frac{1}{12} y^{2} + C_{1} = \int x^{3 \cdot - 1} d x$

Step 2.3.1.2.2

Multiply $3$ by $- 1$ .

$\frac{1}{12} y^{2} + C_{1} = \int x^{- 3} d x$

Step 2.3.2

By the Power Rule, the integral of $x^{- 3}$ with respect to $x$ is $- \frac{1}{2} x^{- 2}$ .

$\frac{1}{12} y^{2} + C_{1} = - \frac{1}{2} x^{- 2} + C_{2}$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $- \frac{1}{2} x^{- 2} + C_{2}$ as $- \frac{1}{2} \cdot \frac{1}{x^{2}} + C_{2}$ .

$\frac{1}{12} y^{2} + C_{1} = - \frac{1}{2} \cdot \frac{1}{x^{2}} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

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

$\frac{1}{12} y^{2} + C_{1} = - \frac{1}{x^{2} \cdot 2} + C_{2}$

Step 2.3.3.2.2

Move $2$ to the left of $x^{2}$ .

$\frac{1}{12} y^{2} + C_{1} = - \frac{1}{2 x^{2}} + C_{2}$

Step 2.4

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

$\frac{1}{12} y^{2} = - \frac{1}{2 x^{2}} + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $12$ .

$12 (\frac{1}{12} y^{2}) = 12 (- \frac{1}{2 x^{2}} + K)$

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 $12 (\frac{1}{12} y^{2})$ .

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Step 3.2.1.1.1

Combine $\frac{1}{12}$ and $y^{2}$ .

$12 \frac{y^{2}}{12} = 12 (- \frac{1}{2 x^{2}} + K)$

Step 3.2.1.1.2

Cancel the common factor of $12$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$12 \frac{y^{2}}{12} = 12 (- \frac{1}{2 x^{2}} + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 12 (- \frac{1}{2 x^{2}} + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $12 (- \frac{1}{2 x^{2}} + K)$ .

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Step 3.2.2.1.1

Apply the distributive property.

$y^{2} = 12 (- \frac{1}{2 x^{2}}) + 12 K$

Step 3.2.2.1.2

Cancel the common factor of $2$ .

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Step 3.2.2.1.2.1

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

$y^{2} = 12 \frac{- 1}{2 x^{2}} + 12 K$

Step 3.2.2.1.2.2

Factor $2$ out of $12$ .

$y^{2} = 2 (6) \frac{- 1}{2 x^{2}} + 12 K$

Step 3.2.2.1.2.3

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

$y^{2} = 2 (6) \frac{- 1}{2 (x^{2})} + 12 K$

Step 3.2.2.1.2.4

Cancel the common factor.

$y^{2} = 2 \cdot 6 \frac{- 1}{2 x^{2}} + 12 K$

Step 3.2.2.1.2.5

Rewrite the expression.

$y^{2} = 6 \frac{- 1}{x^{2}} + 12 K$

Step 3.2.2.1.3

Combine $6$ and $\frac{- 1}{x^{2}}$ .

$y^{2} = \frac{6 \cdot - 1}{x^{2}} + 12 K$

Step 3.2.2.1.4

Simplify the expression.

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Step 3.2.2.1.4.1

Multiply $6$ by $- 1$ .

$y^{2} = \frac{- 6}{x^{2}} + 12 K$

Step 3.2.2.1.4.2

Move the negative in front of the fraction.

$y^{2} = - \frac{6}{x^{2}} + 12 K$

Step 3.3

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

$y = \pm \sqrt{- \frac{6}{x^{2}} + 12 K}$

Step 3.4

Simplify $\pm \sqrt{- \frac{6}{x^{2}} + 12 K}$ .

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Step 3.4.1

Factor $6$ out of $- \frac{6}{x^{2}} + 12 K$ .

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Step 3.4.1.1

Factor $6$ out of $- \frac{6}{x^{2}}$ .

$y = \pm \sqrt{6 (- \frac{1}{x^{2}}) + 12 K}$

Step 3.4.1.2

Factor $6$ out of $12 K$ .

$y = \pm \sqrt{6 (- \frac{1}{x^{2}}) + 6 (2 K)}$

Step 3.4.1.3

Factor $6$ out of $6 (- \frac{1}{x^{2}}) + 6 (2 K)$ .

$y = \pm \sqrt{6 (- \frac{1}{x^{2}} + 2 K)}$

Step 3.4.2

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

$y = \pm \sqrt{6 (- \frac{1}{x^{2}} + \frac{2 K x^{2}}{x^{2}})}$

Step 3.4.3

Combine the numerators over the common denominator.

$y = \pm \sqrt{6 \frac{- 1 + 2 K x^{2}}{x^{2}}}$

Step 3.4.4

Combine $6$ and $\frac{- 1 + 2 K x^{2}}{x^{2}}$ .

$y = \pm \sqrt{\frac{6 (- 1 + 2 K x^{2})}{x^{2}}}$

Step 3.4.5

Rewrite $\frac{6 (- 1 + 2 K x^{2})}{x^{2}}$ as ${(\frac{1}{x})}^{2} (6 (- 1 + 2 K x^{2}))$ .

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Step 3.4.5.1

Factor the perfect power $1^{2}$ out of $6 (- 1 + 2 K x^{2})$ .

$y = \pm \sqrt{\frac{1^{2} (6 (- 1 + 2 K x^{2}))}{x^{2}}}$

Step 3.4.5.2

Factor the perfect power $x^{2}$ out of $x^{2}$ .

$y = \pm \sqrt{\frac{1^{2} (6 (- 1 + 2 K x^{2}))}{x^{2} \cdot 1}}$

Step 3.4.5.3

Rearrange the fraction $\frac{1^{2} (6 (- 1 + 2 K x^{2}))}{x^{2} \cdot 1}$ .

$y = \pm \sqrt{{(\frac{1}{x})}^{2} (6 (- 1 + 2 K x^{2}))}$

Step 3.4.6

Pull terms out from under the radical.

$y = \pm \frac{1}{x} \sqrt{6 (- 1 + 2 K x^{2})}$

Step 3.4.7

Combine $\frac{1}{x}$ and $\sqrt{6 (- 1 + 2 K x^{2})}$ .

$y = \pm \frac{\sqrt{6 (- 1 + 2 K x^{2})}}{x}$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{\sqrt{6 (- 1 + 2 K x^{2})}}{x}$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{\sqrt{6 (- 1 + 2 K x^{2})}}{x}$

Step 3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{\sqrt{6 (- 1 + 2 K x^{2})}}{x}$

$y = - \frac{\sqrt{6 (- 1 + 2 K x^{2})}}{x}$

$y = \frac{\sqrt{6 (- 1 + 2 K x^{2})}}{x}$

$y = - \frac{\sqrt{6 (- 1 + 2 K x^{2})}}{x}$

$y = \frac{\sqrt{6 (- 1 + 2 K x^{2})}}{x}$

$y = - \frac{\sqrt{6 (- 1 + 2 K x^{2})}}{x}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt{6 (- 1 + K x^{2})}}{x}$

$y = - \frac{\sqrt{6 (- 1 + K x^{2})}}{x}$