Calculus Examples

$x^{3} + 2 y \frac{d y}{d x} = 0$

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

Subtract $x^{3}$ from both sides of the equation.

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

Step 1.1.2

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

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Step 1.1.2.1

Divide each term in $2 y \frac{d y}{d x} = - x^{3}$ by $2 y$ .

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

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $2$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

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

Step 1.1.2.2.1.2

Rewrite the expression.

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

Step 1.1.2.2.2

Cancel the common factor of $y$ .

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Step 1.1.2.2.2.1

Cancel the common factor.

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

Step 1.1.2.2.2.2

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

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

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Move the negative in front of the fraction.

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

Step 1.2

Multiply both sides by $y$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.2

Cancel the common factor of $y$ .

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Step 1.3.2.1

Factor $y$ out of $- y$ .

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

Step 1.3.2.2

Factor $y$ out of $2 y$ .

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

Step 1.3.2.3

Cancel the common factor.

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

Step 1.3.2.4

Rewrite the expression.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{2} (\frac{1}{4} x^{4} + C_{2})$

Step 2.3.4

Simplify the answer.

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Step 2.3.4.1

Rewrite $- \frac{1}{2} (\frac{1}{4} x^{4} + C_{2})$ as $- \frac{1}{2} \cdot \frac{1}{4} x^{4} + C_{2}$ .

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{2} \cdot \frac{1}{4} x^{4} + C_{2}$

Step 2.3.4.2

Simplify.

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Step 2.3.4.2.1

Multiply $\frac{1}{4}$ by $\frac{1}{2}$ .

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{4 \cdot 2} x^{4} + C_{2}$

Step 2.3.4.2.2

Multiply $4$ by $2$ .

$\frac{1}{2} y^{2} + C_{1} = - \frac{1}{8} x^{4} + C_{2}$

Step 2.4

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

$\frac{1}{2} y^{2} = - \frac{1}{8} x^{4} + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (- \frac{1}{8} x^{4} + K)$

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

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Step 3.2.1.1.1

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

$2 \frac{y^{2}}{2} = 2 (- \frac{1}{8} x^{4} + K)$

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.

$2 \frac{y^{2}}{2} = 2 (- \frac{1}{8} x^{4} + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (- \frac{1}{8} x^{4} + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (- \frac{1}{8} x^{4} + K)$ .

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Step 3.2.2.1.1

Combine $x^{4}$ and $\frac{1}{8}$ .

$y^{2} = 2 (- \frac{x^{4}}{8} + K)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{2} = 2 (- \frac{x^{4}}{8}) + 2 K$

Step 3.2.2.1.3

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.1

Move the leading negative in $- \frac{x^{4}}{8}$ into the numerator.

$y^{2} = 2 \frac{- x^{4}}{8} + 2 K$

Step 3.2.2.1.3.2

Factor $2$ out of $8$ .

$y^{2} = 2 \frac{- x^{4}}{2 (4)} + 2 K$

Step 3.2.2.1.3.3

Cancel the common factor.

$y^{2} = 2 \frac{- x^{4}}{2 \cdot 4} + 2 K$

Step 3.2.2.1.3.4

Rewrite the expression.

$y^{2} = \frac{- x^{4}}{4} + 2 K$

Step 3.2.2.1.4

Move the negative in front of the fraction.

$y^{2} = - \frac{x^{4}}{4} + 2 K$

Step 3.3

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

$y = \pm \sqrt{- \frac{x^{4}}{4} + 2 K}$

Step 3.4

Simplify $\pm \sqrt{- \frac{x^{4}}{4} + 2 K}$ .

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Step 3.4.1

To write $2 K$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \pm \sqrt{- \frac{x^{4}}{4} + 2 K \cdot \frac{4}{4}}$

Step 3.4.2

Combine $2 K$ and $\frac{4}{4}$ .

$y = \pm \sqrt{- \frac{x^{4}}{4} + \frac{2 K \cdot 4}{4}}$

Step 3.4.3

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{- x^{4} + 2 K \cdot 4}{4}}$

Step 3.4.4

Multiply $4$ by $2$ .

$y = \pm \sqrt{\frac{- x^{4} + 8 K}{4}}$

Step 3.4.5

Rewrite $\frac{- x^{4} + 8 K}{4}$ as ${(\frac{1}{2})}^{2} (- x^{4} + 8 K)$ .

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Step 3.4.5.1

Factor the perfect power $1^{2}$ out of $- x^{4} + 8 K$ .

$y = \pm \sqrt{\frac{1^{2} (- x^{4} + 8 K)}{4}}$

Step 3.4.5.2

Factor the perfect power $2^{2}$ out of $4$ .

$y = \pm \sqrt{\frac{1^{2} (- x^{4} + 8 K)}{2^{2} \cdot 1}}$

Step 3.4.5.3

Rearrange the fraction $\frac{1^{2} (- x^{4} + 8 K)}{2^{2} \cdot 1}$ .

$y = \pm \sqrt{{(\frac{1}{2})}^{2} (- x^{4} + 8 K)}$

Step 3.4.6

Pull terms out from under the radical.

$y = \pm \frac{1}{2} \sqrt{- x^{4} + 8 K}$

Step 3.4.7

Combine $\frac{1}{2}$ and $\sqrt{- x^{4} + 8 K}$ .

$y = \pm \frac{\sqrt{- x^{4} + 8 K}}{2}$

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.

$y = \frac{\sqrt{- x^{4} + 8 K}}{2}$

Step 3.5.2

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

$y = - \frac{\sqrt{- x^{4} + 8 K}}{2}$

Step 3.5.3

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

$y = \frac{\sqrt{- x^{4} + 8 K}}{2}$

$y = - \frac{\sqrt{- x^{4} + 8 K}}{2}$

$y = \frac{\sqrt{- x^{4} + 8 K}}{2}$

$y = - \frac{\sqrt{- x^{4} + 8 K}}{2}$

$y = \frac{\sqrt{- x^{4} + 8 K}}{2}$

$y = - \frac{\sqrt{- x^{4} + 8 K}}{2}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt{- x^{4} + K}}{2}$

$y = - \frac{\sqrt{- x^{4} + K}}{2}$