Calculus Examples

$\frac{1}{y} d x + 3 y d y = 0$

Step 1

Subtract $\frac{1}{y} d x$ from both sides of the equation.

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

Step 2

Multiply both sides by $y$ .

$y (3 y) d y = y (- \frac{1}{y}) d x$

Step 3

Simplify.

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Step 3.1

Rewrite using the commutative property of multiplication.

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

Step 3.2

Multiply $y$ by $y$ by adding the exponents.

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Step 3.2.1

Move $y$ .

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

Step 3.2.2

Multiply $y$ by $y$ .

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

Cancel the common factor of $y$ .

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Step 3.4.1

Factor $y$ out of $- y$ .

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

Since $3$ is constant with respect to $y$ , move $3$ out of the integral.

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

Step 4.2.2

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

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

Step 4.2.3

Simplify the answer.

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Step 4.2.3.1

Rewrite $3 (\frac{1}{3} y^{3} + C_{1})$ as $3 (\frac{1}{3}) y^{3} + C_{1}$ .

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

Step 4.2.3.2

Simplify.

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Step 4.2.3.2.1

Combine $3$ and $\frac{1}{3}$ .

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

Step 4.2.3.2.2

Cancel the common factor of $3$ .

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Step 4.2.3.2.2.1

Cancel the common factor.

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

Step 4.2.3.2.2.2

Rewrite the expression.

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

Step 4.2.3.2.3

Multiply $y^{3}$ by $1$ .

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

Step 4.3

Apply the constant rule.

$y^{3} + C_{1} = - x + C_{2}$

Step 4.4

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

$y^{3} = - x + K$

Step 5

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

$y = \sqrt[3]{- x + K}$