Calculus Examples

$\frac{d y}{d x} + 12 x y^{4} = 0$

Step 1

Separate the variables.

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Step 1.1

Subtract $12 x y^{4}$ from both sides of the equation.

$\frac{d y}{d x} = - 12 x y^{4}$

Step 1.2

Multiply both sides by $\frac{1}{y^{4}}$ .

$\frac{1}{y^{4}} \frac{d y}{d x} = \frac{1}{y^{4}} (- 12 x y^{4})$

Step 1.3

Simplify.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

$\frac{1}{y^{4}} \frac{d y}{d x} = - 12 \frac{1}{y^{4}} (x y^{4})$

Step 1.3.2

Combine $- 12$ and $\frac{1}{y^{4}}$ .

$\frac{1}{y^{4}} \frac{d y}{d x} = \frac{- 12}{y^{4}} (x y^{4})$

Step 1.3.3

Cancel the common factor of $y^{4}$ .

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Step 1.3.3.1

Factor $y^{4}$ out of $x y^{4}$ .

$\frac{1}{y^{4}} \frac{d y}{d x} = \frac{- 12}{y^{4}} (y^{4} x)$

Step 1.3.3.2

Cancel the common factor.

$\frac{1}{y^{4}} \frac{d y}{d x} = \frac{- 12}{y^{4}} (y^{4} x)$

Step 1.3.3.3

Rewrite the expression.

$\frac{1}{y^{4}} \frac{d y}{d x} = - 12 x$

Step 1.4

Rewrite the equation.

$\frac{1}{y^{4}} d y = - 12 x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y^{4}} d y = \int - 12 x d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply basic rules of exponents.

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Step 2.2.1.1

Move $y^{4}$ out of the denominator by raising it to the $- 1$ power.

$\int {(y^{4})}^{- 1} d y = \int - 12 x d x$

Step 2.2.1.2

Multiply the exponents in ${(y^{4})}^{- 1}$ .

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Step 2.2.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int y^{4 \cdot - 1} d y = \int - 12 x d x$

Step 2.2.1.2.2

Multiply $4$ by $- 1$ .

$\int y^{- 4} d y = \int - 12 x d x$

Step 2.2.2

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

$- \frac{1}{3} y^{- 3} + C_{1} = \int - 12 x d x$

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

Rewrite $- \frac{1}{3} y^{- 3} + C_{1}$ as $- \frac{1}{3} \cdot \frac{1}{y^{3}} + C_{1}$ .

$- \frac{1}{3} \cdot \frac{1}{y^{3}} + C_{1} = \int - 12 x d x$

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

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

$- \frac{1}{y^{3} \cdot 3} + C_{1} = \int - 12 x d x$

Step 2.2.3.2.2

Move $3$ to the left of $y^{3}$ .

$- \frac{1}{3 y^{3}} + C_{1} = \int - 12 x d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $- 12$ is constant with respect to $x$ , move $- 12$ out of the integral.

$- \frac{1}{3 y^{3}} + C_{1} = - 12 \int x d x$

Step 2.3.2

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

$- \frac{1}{3 y^{3}} + C_{1} = - 12 (\frac{1}{2} x^{2} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $- 12 (\frac{1}{2} x^{2} + C_{2})$ as $- 12 (\frac{1}{2}) x^{2} + C_{2}$ .

$- \frac{1}{3 y^{3}} + C_{1} = - 12 (\frac{1}{2}) x^{2} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $- 12$ and $\frac{1}{2}$ .

$- \frac{1}{3 y^{3}} + C_{1} = \frac{- 12}{2} x^{2} + C_{2}$

Step 2.3.3.2.2

Cancel the common factor of $- 12$ and $2$ .

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Step 2.3.3.2.2.1

Factor $2$ out of $- 12$ .

$- \frac{1}{3 y^{3}} + C_{1} = \frac{2 \cdot - 6}{2} x^{2} + C_{2}$

Step 2.3.3.2.2.2

Cancel the common factors.

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Step 2.3.3.2.2.2.1

Factor $2$ out of $2$ .

$- \frac{1}{3 y^{3}} + C_{1} = \frac{2 \cdot - 6}{2 (1)} x^{2} + C_{2}$

Step 2.3.3.2.2.2.2

Cancel the common factor.

$- \frac{1}{3 y^{3}} + C_{1} = \frac{2 \cdot - 6}{2 \cdot 1} x^{2} + C_{2}$

Step 2.3.3.2.2.2.3

Rewrite the expression.

$- \frac{1}{3 y^{3}} + C_{1} = \frac{- 6}{1} x^{2} + C_{2}$

Step 2.3.3.2.2.2.4

Divide $- 6$ by $1$ .

$- \frac{1}{3 y^{3}} + C_{1} = - 6 x^{2} + C_{2}$

Step 2.4

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

$- \frac{1}{3 y^{3}} = - 6 x^{2} + K$

Step 3

Solve for $y$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$3 y^{3}, 1, 1$

Step 3.1.2

The LCM of one and any expression is the expression.

$3 y^{3}$

Step 3.2

Multiply each term in $- \frac{1}{3 y^{3}} = - 6 x^{2} + K$ by $3 y^{3}$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $- \frac{1}{3 y^{3}} = - 6 x^{2} + K$ by $3 y^{3}$ .

$- \frac{1}{3 y^{3}} (3 y^{3}) = - 6 x^{2} (3 y^{3}) + K (3 y^{3})$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $3 y^{3}$ .

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Step 3.2.2.1.1

Move the leading negative in $- \frac{1}{3 y^{3}}$ into the numerator.

$\frac{- 1}{3 y^{3}} (3 y^{3}) = - 6 x^{2} (3 y^{3}) + K (3 y^{3})$

Step 3.2.2.1.2

Cancel the common factor.

$\frac{- 1}{3 y^{3}} (3 y^{3}) = - 6 x^{2} (3 y^{3}) + K (3 y^{3})$

Step 3.2.2.1.3

Rewrite the expression.

$- 1 = - 6 x^{2} (3 y^{3}) + K (3 y^{3})$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Rewrite using the commutative property of multiplication.

$- 1 = - 6 \cdot 3 x^{2} y^{3} + K (3 y^{3})$

Step 3.2.3.1.2

Multiply $- 6$ by $3$ .

$- 1 = - 18 x^{2} y^{3} + K (3 y^{3})$

Step 3.2.3.1.3

Rewrite using the commutative property of multiplication.

$- 1 = - 18 x^{2} y^{3} + 3 K y^{3}$

Step 3.3

Solve the equation.

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Step 3.3.1

Rewrite the equation as $- 18 x^{2} y^{3} + 3 K y^{3} = - 1$ .

$- 18 x^{2} y^{3} + 3 K y^{3} = - 1$

Step 3.3.2

Factor $3 y^{3}$ out of $- 18 x^{2} y^{3} + 3 K y^{3}$ .

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Step 3.3.2.1

Factor $3 y^{3}$ out of $- 18 x^{2} y^{3}$ .

$3 y^{3} (- 6 x^{2}) + 3 K y^{3} = - 1$

Step 3.3.2.2

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

$3 y^{3} (- 6 x^{2}) + 3 y^{3} K = - 1$

Step 3.3.2.3

Factor $3 y^{3}$ out of $3 y^{3} (- 6 x^{2}) + 3 y^{3} K$ .

$3 y^{3} (- 6 x^{2} + K) = - 1$

Step 3.3.3

Divide each term in $3 y^{3} (- 6 x^{2} + K) = - 1$ by $3 (- 6 x^{2} + K)$ and simplify.

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Step 3.3.3.1

Divide each term in $3 y^{3} (- 6 x^{2} + K) = - 1$ by $3 (- 6 x^{2} + K)$ .

$\frac{3 y^{3} (- 6 x^{2} + K)}{3 (- 6 x^{2} + K)} = \frac{- 1}{3 (- 6 x^{2} + K)}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $3$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{3 y^{3} (- 6 x^{2} + K)}{3 (- 6 x^{2} + K)} = \frac{- 1}{3 (- 6 x^{2} + K)}$

Step 3.3.3.2.1.2

Rewrite the expression.

$\frac{y^{3} (- 6 x^{2} + K)}{- 6 x^{2} + K} = \frac{- 1}{3 (- 6 x^{2} + K)}$

Step 3.3.3.2.2

Cancel the common factor of $- 6 x^{2} + K$ .

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Step 3.3.3.2.2.1

Cancel the common factor.

$\frac{y^{3} (- 6 x^{2} + K)}{- 6 x^{2} + K} = \frac{- 1}{3 (- 6 x^{2} + K)}$

Step 3.3.3.2.2.2

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

$y^{3} = \frac{- 1}{3 (- 6 x^{2} + K)}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Move the negative in front of the fraction.

$y^{3} = - \frac{1}{3 (- 6 x^{2} + K)}$

Step 3.3.3.3.2

Factor $- 1$ out of $- 6 x^{2}$ .

$y^{3} = - \frac{1}{3 (- (6 x^{2}) + K)}$

Step 3.3.3.3.3

Factor $- 1$ out of $K$ .

$y^{3} = - \frac{1}{3 (- (6 x^{2}) - 1 (- K))}$

Step 3.3.3.3.4

Factor $- 1$ out of $- (6 x^{2}) - 1 (- K)$ .

$y^{3} = - \frac{1}{3 (- (6 x^{2} - K))}$

Step 3.3.3.3.5

Simplify the expression.

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Step 3.3.3.3.5.1

Rewrite $- (6 x^{2} - K)$ as $- 1 (6 x^{2} - K)$ .

$y^{3} = - \frac{1}{3 (- 1 (6 x^{2} - K))}$

Step 3.3.3.3.5.2

Move the negative in front of the fraction.

$y^{3} = - - \frac{1}{3 (6 x^{2} - K)}$

Step 3.3.3.3.5.3

Multiply $- 1$ by $- 1$ .

$y^{3} = 1 \frac{1}{3 (6 x^{2} - K)}$

Step 3.3.3.3.5.4

Multiply $\frac{1}{3 (6 x^{2} - K)}$ by $1$ .

$y^{3} = \frac{1}{3 (6 x^{2} - K)}$

Step 3.3.4

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

$y = \sqrt[3]{\frac{1}{3 (6 x^{2} - K)}}$

Step 3.3.5

Simplify $\sqrt[3]{\frac{1}{3 (6 x^{2} - K)}}$ .

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Step 3.3.5.1

Rewrite $\sqrt[3]{\frac{1}{3 (6 x^{2} - K)}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{3 (6 x^{2} - K)}}$ .

$y = \frac{\sqrt[3]{1}}{\sqrt[3]{3 (6 x^{2} - K)}}$

Step 3.3.5.2

Any root of $1$ is $1$ .

$y = \frac{1}{\sqrt[3]{3 (6 x^{2} - K)}}$

Step 3.3.5.3

Multiply $\frac{1}{\sqrt[3]{3 (6 x^{2} - K)}}$ by $\frac{{\sqrt[3]{3 (6 x^{2} - K)}}^{2}}{{\sqrt[3]{3 (6 x^{2} - K)}}^{2}}$ .

$y = \frac{1}{\sqrt[3]{3 (6 x^{2} - K)}} \cdot \frac{{\sqrt[3]{3 (6 x^{2} - K)}}^{2}}{{\sqrt[3]{3 (6 x^{2} - K)}}^{2}}$

Step 3.3.5.4

Combine and simplify the denominator.

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Step 3.3.5.4.1

Multiply $\frac{1}{\sqrt[3]{3 (6 x^{2} - K)}}$ by $\frac{{\sqrt[3]{3 (6 x^{2} - K)}}^{2}}{{\sqrt[3]{3 (6 x^{2} - K)}}^{2}}$ .

$y = \frac{{\sqrt[3]{3 (6 x^{2} - K)}}^{2}}{\sqrt[3]{3 (6 x^{2} - K)} {\sqrt[3]{3 (6 x^{2} - K)}}^{2}}$

Step 3.3.5.4.2

Raise $\sqrt[3]{3 (6 x^{2} - K)}$ to the power of $1$ .

$y = \frac{{\sqrt[3]{3 (6 x^{2} - K)}}^{2}}{{\sqrt[3]{3 (6 x^{2} - K)}}^{1} {\sqrt[3]{3 (6 x^{2} - K)}}^{2}}$

Step 3.3.5.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{{\sqrt[3]{3 (6 x^{2} - K)}}^{2}}{{\sqrt[3]{3 (6 x^{2} - K)}}^{1 + 2}}$

Step 3.3.5.4.4

Add $1$ and $2$ .

$y = \frac{{\sqrt[3]{3 (6 x^{2} - K)}}^{2}}{{\sqrt[3]{3 (6 x^{2} - K)}}^{3}}$

Step 3.3.5.4.5

Rewrite ${\sqrt[3]{3 (6 x^{2} - K)}}^{3}$ as $3 (6 x^{2} - K)$ .

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Step 3.3.5.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{3 (6 x^{2} - K)}$ as ${(3 (6 x^{2} - K))}^{\frac{1}{3}}$ .

$y = \frac{{\sqrt[3]{3 (6 x^{2} - K)}}^{2}}{{({(3 (6 x^{2} - K))}^{\frac{1}{3}})}^{3}}$

Step 3.3.5.4.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \frac{{\sqrt[3]{3 (6 x^{2} - K)}}^{2}}{{(3 (6 x^{2} - K))}^{\frac{1}{3} \cdot 3}}$

Step 3.3.5.4.5.3

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

$y = \frac{{\sqrt[3]{3 (6 x^{2} - K)}}^{2}}{{(3 (6 x^{2} - K))}^{\frac{3}{3}}}$

Step 3.3.5.4.5.4

Cancel the common factor of $3$ .

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Step 3.3.5.4.5.4.1

Cancel the common factor.

$y = \frac{{\sqrt[3]{3 (6 x^{2} - K)}}^{2}}{{(3 (6 x^{2} - K))}^{\frac{3}{3}}}$

Step 3.3.5.4.5.4.2

Rewrite the expression.

$y = \frac{{\sqrt[3]{3 (6 x^{2} - K)}}^{2}}{{(3 (6 x^{2} - K))}^{1}}$

Step 3.3.5.4.5.5

Simplify.

$y = \frac{{\sqrt[3]{3 (6 x^{2} - K)}}^{2}}{3 (6 x^{2} - K)}$

Step 3.3.5.5

Simplify the numerator.

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Step 3.3.5.5.1

Rewrite ${\sqrt[3]{3 (6 x^{2} - K)}}^{2}$ as $\sqrt[3]{{(3 (6 x^{2} - K))}^{2}}$ .

$y = \frac{\sqrt[3]{{(3 (6 x^{2} - K))}^{2}}}{3 (6 x^{2} - K)}$

Step 3.3.5.5.2

Apply the product rule to $3 (6 x^{2} - K)$ .

$y = \frac{\sqrt[3]{3^{2} {(6 x^{2} - K)}^{2}}}{3 (6 x^{2} - K)}$

Step 3.3.5.5.3

Raise $3$ to the power of $2$ .

$y = \frac{\sqrt[3]{9 {(6 x^{2} - K)}^{2}}}{3 (6 x^{2} - K)}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt[3]{9 {(6 x^{2} + K)}^{2}}}{3 (6 x^{2} + K)}$