Calculus Examples

$x d x - 4 y d y = 0$

Step 1

Subtract $x d x$ from both sides of the equation.

$- 4 y d y = - x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int - 4 y d y = \int - x d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Since $- 4$ is constant with respect to $y$ , move $- 4$ out of the integral.

$- 4 \int y d y = \int - x d x$

Step 2.2.2

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

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

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

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

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

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

Combine $- 4$ and $\frac{1}{2}$ .

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

Step 2.2.3.2.2

Cancel the common factor of $- 4$ and $2$ .

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Step 2.2.3.2.2.1

Factor $2$ out of $- 4$ .

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

Step 2.2.3.2.2.2

Cancel the common factors.

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Step 2.2.3.2.2.2.1

Factor $2$ out of $2$ .

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

Step 2.2.3.2.2.2.2

Cancel the common factor.

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

Step 2.2.3.2.2.2.3

Rewrite the expression.

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

Step 2.2.3.2.2.2.4

Divide $- 2$ by $1$ .

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

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.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $- 2 y^{2} = - \frac{1}{2} x^{2} + K$ by $- 2$ and simplify.

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Step 3.1.1

Divide each term in $- 2 y^{2} = - \frac{1}{2} x^{2} + K$ by $- 2$ .

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $- 2$ .

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Step 3.1.2.1.1

Cancel the common factor.

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

Step 3.1.2.1.2

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

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

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

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

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

Step 3.1.3.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.1.3.1.3

Move the negative in front of the fraction.

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

Step 3.1.3.1.4

Multiply $- \frac{x^{2}}{2} (- \frac{1}{2})$ .

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Step 3.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.3.1.4.2

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

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

Step 3.1.3.1.4.3

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

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

Step 3.1.3.1.4.4

Multiply $2$ by $2$ .

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

Step 3.1.3.1.5

Move the negative in front of the fraction.

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.2.1

Multiply $\frac{K}{2}$ by $\frac{2}{2}$ .

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

Step 3.3.2.2

Multiply $2$ by $2$ .

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

Step 3.3.3

Combine the numerators over the common denominator.

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

Step 3.3.4

Multiply $2$ by $- 1$ .

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

Step 3.3.5

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

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Step 3.3.5.1

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

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

Step 3.3.5.2

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

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

Step 3.3.5.3

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

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

Step 3.3.6

Pull terms out from under the radical.

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

Step 3.3.7

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

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

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

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

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

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

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

Step 4

Simplify the constant of integration.

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

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