Calculus Examples

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

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

Step 1

Subtract

\frac{1}{y} d x

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

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

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

Step 2

Multiply both sides by

y

$y$ .

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

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Rewrite using the commutative property of multiplication.

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

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

Step 3.2

Multiply

y

$y$ by

y

$y$ by adding the exponents.

Tap for more steps...

Step 3.2.1

Move

y

$y$ .

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

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

Step 3.2.2

Multiply

y

$y$ by

y

$y$ .

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

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

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

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

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

Step 3.4

Cancel the common factor of

y

$y$ .

Tap for more steps...

Step 3.4.1

Factor

y

$y$ out of

- y

$- y$ .

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

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

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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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

$3$ .

Tap for more steps...

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.

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