Calculus Examples

$2 x y d x + y^{2} d y = 0$

Step 1

Subtract $2 x y d x$ from both sides of the equation.

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

Step 2

Multiply both sides by $\frac{1}{y}$ .

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $y$ .

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

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

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

Step 3.4

Cancel the common factor of $y$ .

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Step 3.4.1

Factor $y$ out of $x y$ .

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

$y d y = - 2 x d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int y d y = \int - 2 x d x$

Step 4.2

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

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

Step 4.3

Integrate the right side.

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Step 4.3.1

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

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

Step 4.3.2

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

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

Step 4.3.3

Simplify the answer.

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Step 4.3.3.1

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

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

Step 4.3.3.2

Simplify.

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Step 4.3.3.2.1

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

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

Step 4.3.3.2.2

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

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Step 4.3.3.2.2.1

Factor $2$ out of $- 2$ .

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

Step 4.3.3.2.2.2

Cancel the common factors.

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Step 4.3.3.2.2.2.1

Factor $2$ out of $2$ .

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

Step 4.3.3.2.2.2.2

Cancel the common factor.

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

Step 4.3.3.2.2.2.3

Rewrite the expression.

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

Step 4.3.3.2.2.2.4

Divide $- 1$ by $1$ .

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

Step 4.4

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

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

Step 5

Solve for $y$ .

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Step 5.1

Multiply both sides of the equation by $2$ .

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

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

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

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Step 5.2.1.1.1

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

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

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

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

Step 5.2.1.1.2.2

Rewrite the expression.

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

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

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Step 5.2.2.1.1

Apply the distributive property.

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

Step 5.2.2.1.2

Multiply $- 1$ by $2$ .

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

Step 5.3

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

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

Step 5.4

Factor $2$ out of $- 2 x^{2} + 2 K$ .

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Step 5.4.1

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

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

Step 5.4.2

Factor $2$ out of $2 K$ .

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

Step 5.4.3

Factor $2$ out of $2 (- x^{2}) + 2 K$ .

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

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

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

Step 5.5.3

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

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

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

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

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

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

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