Calculus Examples

$x d x - y d y = 0$

Step 1

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

$- y d y = - x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

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.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

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

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Step 3.2.1.1.1

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

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

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 $- \frac{y^{2}}{2}$ into the numerator.

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

Step 3.2.1.1.2.2

Factor $2$ out of $- 2$ .

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

Step 3.2.1.1.2.3

Cancel the common factor.

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

Step 3.2.1.1.2.4

Rewrite the expression.

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

Step 3.2.1.1.3

Multiply.

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Step 3.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2.1.1.3.2

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

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

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

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

Step 3.2.2.1.2

Apply the distributive property.

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

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 $- \frac{x^{2}}{2}$ into the numerator.

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

Step 3.2.2.1.3.2

Factor $2$ out of $- 2$ .

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

Step 3.2.2.1.3.3

Cancel the common factor.

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

Step 3.2.2.1.3.4

Rewrite the expression.

$y^{2} = - - x^{2} - 2 K$

Step 3.2.2.1.4

Multiply.

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Step 3.2.2.1.4.1

Multiply $- 1$ by $- 1$ .

$y^{2} = 1 x^{2} - 2 K$

Step 3.2.2.1.4.2

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

$y^{2} = x^{2} - 2 K$

Step 3.3

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

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

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.

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

Step 3.4.2

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

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

Step 3.4.3

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

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

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

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

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

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

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

Step 4

Simplify the constant of integration.

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

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