Calculus Examples

$d x - 4 x y^{3} d y = 0$

Step 1

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

$- 4 x y^{3} d y = - d x$

Step 2

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

$\frac{1}{x} (- 4 x y^{3}) d y = \frac{1}{x} \cdot - 1 d x$

Step 3

Simplify.

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Step 3.1

Rewrite using the commutative property of multiplication.

$- 4 \frac{1}{x} (x y^{3}) d y = \frac{1}{x} \cdot - 1 d x$

Step 3.2

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

$\frac{- 4}{x} (x y^{3}) d y = \frac{1}{x} \cdot - 1 d x$

Step 3.3

Cancel the common factor of $x$ .

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Step 3.3.1

Factor $x$ out of $x y^{3}$ .

$\frac{- 4}{x} (x (y^{3})) d y = \frac{1}{x} \cdot - 1 d x$

Step 3.3.2

Cancel the common factor.

$\frac{- 4}{x} (x y^{3}) d y = \frac{1}{x} \cdot - 1 d x$

Step 3.3.3

Rewrite the expression.

$- 4 y^{3} d y = \frac{1}{x} \cdot - 1 d x$

Step 3.4

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

$- 4 y^{3} d y = \frac{- 1}{x} d x$

Step 3.5

Move the negative in front of the fraction.

$- 4 y^{3} d y = - \frac{1}{x} d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int - 4 y^{3} d y = \int - \frac{1}{x} d x$

Step 4.2

Integrate the left side.

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Step 4.2.1

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

$- 4 \int y^{3} d y = \int - \frac{1}{x} d x$

Step 4.2.2

By the Power Rule, the integral of $y^{3}$ with respect to $y$ is $\frac{1}{4} y^{4}$ .

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

Step 4.2.3

Simplify the answer.

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Step 4.2.3.1

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

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

Step 4.2.3.2

Simplify.

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Step 4.2.3.2.1

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

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

Step 4.2.3.2.2

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

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Step 4.2.3.2.2.1

Factor $4$ out of $- 4$ .

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

Step 4.2.3.2.2.2

Cancel the common factors.

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Step 4.2.3.2.2.2.1

Factor $4$ out of $4$ .

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

Step 4.2.3.2.2.2.2

Cancel the common factor.

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

Step 4.2.3.2.2.2.3

Rewrite the expression.

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

Step 4.2.3.2.2.2.4

Divide $- 1$ by $1$ .

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

Step 4.3

Integrate the right side.

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Step 4.3.1

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

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

Step 4.3.2

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$- y^{4} + C_{1} = - (\ln (| x |) + C_{2})$

Step 4.3.3

Simplify.

$- y^{4} + C_{1} = - \ln (| x |) + C_{2}$

Step 4.4

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

$- y^{4} = - \ln (| x |) + C$

Step 5

Solve for $y$ .

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Step 5.1

Divide each term in $- y^{4} = - \ln (| x |) + C$ by $- 1$ and simplify.

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Step 5.1.1

Divide each term in $- y^{4} = - \ln (| x |) + C$ by $- 1$ .

$\frac{- y^{4}}{- 1} = \frac{- \ln (| x |)}{- 1} + \frac{C}{- 1}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Dividing two negative values results in a positive value.

$\frac{y^{4}}{1} = \frac{- \ln (| x |)}{- 1} + \frac{C}{- 1}$

Step 5.1.2.2

Divide $y^{4}$ by $1$ .

$y^{4} = \frac{- \ln (| x |)}{- 1} + \frac{C}{- 1}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Simplify each term.

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Step 5.1.3.1.1

Dividing two negative values results in a positive value.

$y^{4} = \frac{\ln (| x |)}{1} + \frac{C}{- 1}$

Step 5.1.3.1.2

Divide $\ln (| x |)$ by $1$ .

$y^{4} = \ln (| x |) + \frac{C}{- 1}$

Step 5.1.3.1.3

Move the negative one from the denominator of $\frac{C}{- 1}$ .

$y^{4} = \ln (| x |) - 1 \cdot C$

Step 5.1.3.1.4

Rewrite $- 1 \cdot C$ as $- C$ .

$y^{4} = \ln (| x |) - C$

Step 5.2

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

$y = \pm \sqrt[4]{\ln (| x |) - C}$

Step 5.3

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

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Step 5.3.1

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

$y = \sqrt[4]{\ln (| x |) - C}$

Step 5.3.2

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

$y = - \sqrt[4]{\ln (| x |) - C}$

Step 5.3.3

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

$y = \sqrt[4]{\ln (| x |) - C}$

$y = - \sqrt[4]{\ln (| x |) - C}$

$y = \sqrt[4]{\ln (| x |) - C}$

$y = - \sqrt[4]{\ln (| x |) - C}$

$y = \sqrt[4]{\ln (| x |) - C}$

$y = - \sqrt[4]{\ln (| x |) - C}$

Step 6

Simplify the constant of integration.

$y = \sqrt[4]{\ln (| x |) + C}$

$y = - \sqrt[4]{\ln (| x |) + C}$