Calculus Examples

$\frac{d y}{d x} = \frac{6 - x^{2}}{2 y^{3}}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y^{3}$ .

$y^{3} \frac{d y}{d x} = y^{3} \frac{6 - x^{2}}{2 y^{3}}$

Step 1.2

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

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Step 1.2.1

Factor $y^{3}$ out of $2 y^{3}$ .

$y^{3} \frac{d y}{d x} = y^{3} \frac{6 - x^{2}}{y^{3} \cdot 2}$

Step 1.2.2

Cancel the common factor.

$y^{3} \frac{d y}{d x} = y^{3} \frac{6 - x^{2}}{y^{3} \cdot 2}$

Step 1.2.3

Rewrite the expression.

$y^{3} \frac{d y}{d x} = \frac{6 - x^{2}}{2}$

Step 1.3

Rewrite the equation.

$y^{3} d y = \frac{6 - x^{2}}{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y^{3} d y = \int \frac{6 - x^{2}}{2} d x$

Step 2.2

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

$\frac{1}{4} y^{4} + C_{1} = \int \frac{6 - x^{2}}{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{4} y^{4} + C_{1} = \frac{1}{2} \int 6 - x^{2} d x$

Step 2.3.2

Split the single integral into multiple integrals.

$\frac{1}{4} y^{4} + C_{1} = \frac{1}{2} (\int 6 d x + \int - x^{2} d x)$

Step 2.3.3

Apply the constant rule.

$\frac{1}{4} y^{4} + C_{1} = \frac{1}{2} (6 x + C_{2} + \int - x^{2} d x)$

Step 2.3.4

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

$\frac{1}{4} y^{4} + C_{1} = \frac{1}{2} (6 x + C_{2} - \int x^{2} d x)$

Step 2.3.5

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

$\frac{1}{4} y^{4} + C_{1} = \frac{1}{2} (6 x + C_{2} - (\frac{1}{3} x^{3} + C_{3}))$

Step 2.3.6

Simplify.

$\frac{1}{4} y^{4} + C_{1} = \frac{1}{2} (6 x - \frac{1}{3} x^{3}) + C_{4}$

Step 2.4

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

$\frac{1}{4} y^{4} = \frac{1}{2} (6 x - \frac{1}{3} x^{3}) + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $4$ .

$4 (\frac{1}{4} y^{4}) = 4 (\frac{1}{2} (6 x - \frac{1}{3} x^{3}) + K)$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $4 (\frac{1}{4} y^{4})$ .

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Step 3.2.1.1.1

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

$4 \frac{y^{4}}{4} = 4 (\frac{1}{2} (6 x - \frac{1}{3} x^{3}) + K)$

Step 3.2.1.1.2

Cancel the common factor of $4$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$4 \frac{y^{4}}{4} = 4 (\frac{1}{2} (6 x - \frac{1}{3} x^{3}) + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{4} = 4 (\frac{1}{2} (6 x - \frac{1}{3} x^{3}) + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $4 (\frac{1}{2} (6 x - \frac{1}{3} x^{3}) + K)$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

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

$y^{4} = 4 (\frac{1}{2} (6 x - \frac{x^{3}}{3}) + K)$

Step 3.2.2.1.1.2

Apply the distributive property.

$y^{4} = 4 (\frac{1}{2} (6 x) + \frac{1}{2} (- \frac{x^{3}}{3}) + K)$

Step 3.2.2.1.1.3

Cancel the common factor of $2$ .

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Step 3.2.2.1.1.3.1

Factor $2$ out of $6 x$ .

$y^{4} = 4 (\frac{1}{2} (2 (3 x)) + \frac{1}{2} (- \frac{x^{3}}{3}) + K)$

Step 3.2.2.1.1.3.2

Cancel the common factor.

$y^{4} = 4 (\frac{1}{2} (2 (3 x)) + \frac{1}{2} (- \frac{x^{3}}{3}) + K)$

Step 3.2.2.1.1.3.3

Rewrite the expression.

$y^{4} = 4 (3 x + \frac{1}{2} (- \frac{x^{3}}{3}) + K)$

Step 3.2.2.1.1.4

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

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Step 3.2.2.1.1.4.1

Multiply $\frac{1}{2}$ by $\frac{x^{3}}{3}$ .

$y^{4} = 4 (3 x - \frac{x^{3}}{2 \cdot 3} + K)$

Step 3.2.2.1.1.4.2

Multiply $2$ by $3$ .

$y^{4} = 4 (3 x - \frac{x^{3}}{6} + K)$

Step 3.2.2.1.2

Apply the distributive property.

$y^{4} = 4 (3 x) + 4 (- \frac{x^{3}}{6}) + 4 K$

Step 3.2.2.1.3

Simplify.

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Step 3.2.2.1.3.1

Multiply $3$ by $4$ .

$y^{4} = 12 x + 4 (- \frac{x^{3}}{6}) + 4 K$

Step 3.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.2.1

Move the leading negative in $- \frac{x^{3}}{6}$ into the numerator.

$y^{4} = 12 x + 4 \frac{- x^{3}}{6} + 4 K$

Step 3.2.2.1.3.2.2

Factor $2$ out of $4$ .

$y^{4} = 12 x + 2 (2) \frac{- x^{3}}{6} + 4 K$

Step 3.2.2.1.3.2.3

Factor $2$ out of $6$ .

$y^{4} = 12 x + 2 \cdot 2 \frac{- x^{3}}{2 \cdot 3} + 4 K$

Step 3.2.2.1.3.2.4

Cancel the common factor.

$y^{4} = 12 x + 2 \cdot 2 \frac{- x^{3}}{2 \cdot 3} + 4 K$

Step 3.2.2.1.3.2.5

Rewrite the expression.

$y^{4} = 12 x + 2 \frac{- x^{3}}{3} + 4 K$

Step 3.2.2.1.3.3

Combine $2$ and $\frac{- x^{3}}{3}$ .

$y^{4} = 12 x + \frac{2 (- x^{3})}{3} + 4 K$

Step 3.2.2.1.3.4

Multiply $- 1$ by $2$ .

$y^{4} = 12 x + \frac{- 2 x^{3}}{3} + 4 K$

Step 3.2.2.1.4

Move the negative in front of the fraction.

$y^{4} = 12 x - \frac{2 x^{3}}{3} + 4 K$

Step 3.3

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

$y = \pm \sqrt[4]{12 x - \frac{2 x^{3}}{3} + 4 K}$

Step 3.4

Simplify $\pm \sqrt[4]{12 x - \frac{2 x^{3}}{3} + 4 K}$ .

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Step 3.4.1

Factor $2$ out of $12 x - \frac{2 x^{3}}{3} + 4 K$ .

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Step 3.4.1.1

Factor $2$ out of $12 x$ .

$y = \pm \sqrt[4]{2 (6 x) - \frac{2 x^{3}}{3} + 4 K}$

Step 3.4.1.2

Factor $2$ out of $- \frac{2 x^{3}}{3}$ .

$y = \pm \sqrt[4]{2 (6 x) + 2 (- \frac{x^{3}}{3}) + 4 K}$

Step 3.4.1.3

Factor $2$ out of $4 K$ .

$y = \pm \sqrt[4]{2 (6 x) + 2 (- \frac{x^{3}}{3}) + 2 (2 K)}$

Step 3.4.1.4

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

$y = \pm \sqrt[4]{2 (6 x - \frac{x^{3}}{3}) + 2 (2 K)}$

Step 3.4.1.5

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

$y = \pm \sqrt[4]{2 (6 x - \frac{x^{3}}{3} + 2 K)}$

Step 3.4.2

To write $6 x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \pm \sqrt[4]{2 (6 x \cdot \frac{3}{3} - \frac{x^{3}}{3} + 2 K)}$

Step 3.4.3

Simplify terms.

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Step 3.4.3.1

Combine $6 x$ and $\frac{3}{3}$ .

$y = \pm \sqrt[4]{2 (\frac{6 x \cdot 3}{3} - \frac{x^{3}}{3} + 2 K)}$

Step 3.4.3.2

Combine the numerators over the common denominator.

$y = \pm \sqrt[4]{2 (\frac{6 x \cdot 3 - x^{3}}{3} + 2 K)}$

Step 3.4.4

Simplify the numerator.

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Step 3.4.4.1

Factor $x$ out of $6 x \cdot 3 - x^{3}$ .

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Step 3.4.4.1.1

Factor $x$ out of $6 x \cdot 3$ .

$y = \pm \sqrt[4]{2 (\frac{x (6 \cdot 3) - x^{3}}{3} + 2 K)}$

Step 3.4.4.1.2

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

$y = \pm \sqrt[4]{2 (\frac{x (6 \cdot 3) + x (- x^{2})}{3} + 2 K)}$

Step 3.4.4.1.3

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

$y = \pm \sqrt[4]{2 (\frac{x (6 \cdot 3 - x^{2})}{3} + 2 K)}$

Step 3.4.4.2

Multiply $6$ by $3$ .

$y = \pm \sqrt[4]{2 (\frac{x (18 - x^{2})}{3} + 2 K)}$

Step 3.4.5

To write $2 K$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \pm \sqrt[4]{2 (\frac{x (18 - x^{2})}{3} + 2 K \cdot \frac{3}{3})}$

Step 3.4.6

Simplify terms.

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Step 3.4.6.1

Combine $2 K$ and $\frac{3}{3}$ .

$y = \pm \sqrt[4]{2 (\frac{x (18 - x^{2})}{3} + \frac{2 K \cdot 3}{3})}$

Step 3.4.6.2

Combine the numerators over the common denominator.

$y = \pm \sqrt[4]{2 \frac{x (18 - x^{2}) + 2 K \cdot 3}{3}}$

Step 3.4.7

Simplify the numerator.

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Step 3.4.7.1

Apply the distributive property.

$y = \pm \sqrt[4]{2 \frac{x \cdot 18 + x (- x^{2}) + 2 K \cdot 3}{3}}$

Step 3.4.7.2

Move $18$ to the left of $x$ .

$y = \pm \sqrt[4]{2 \frac{18 \cdot x + x (- x^{2}) + 2 K \cdot 3}{3}}$

Step 3.4.7.3

Rewrite using the commutative property of multiplication.

$y = \pm \sqrt[4]{2 \frac{18 \cdot x - x \cdot x^{2} + 2 K \cdot 3}{3}}$

Step 3.4.7.4

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.4.7.4.1

Move $x^{2}$ .

$y = \pm \sqrt[4]{2 \frac{18 x - (x^{2} x) + 2 K \cdot 3}{3}}$

Step 3.4.7.4.2

Multiply $x^{2}$ by $x$ .

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Step 3.4.7.4.2.1

Raise $x$ to the power of $1$ .

$y = \pm \sqrt[4]{2 \frac{18 x - (x^{2} x^{1}) + 2 K \cdot 3}{3}}$

Step 3.4.7.4.2.2

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

$y = \pm \sqrt[4]{2 \frac{18 x - x^{2 + 1} + 2 K \cdot 3}{3}}$

Step 3.4.7.4.3

Add $2$ and $1$ .

$y = \pm \sqrt[4]{2 \frac{18 x - x^{3} + 2 K \cdot 3}{3}}$

Step 3.4.7.5

Multiply $3$ by $2$ .

$y = \pm \sqrt[4]{2 \frac{18 x - x^{3} + 6 K}{3}}$

Step 3.4.8

Combine $2$ and $\frac{18 x - x^{3} + 6 K}{3}$ .

$y = \pm \sqrt[4]{\frac{2 (18 x - x^{3} + 6 K)}{3}}$

Step 3.4.9

Rewrite $\sqrt[4]{\frac{2 (18 x - x^{3} + 6 K)}{3}}$ as $\frac{\sqrt[4]{2 (18 x - x^{3} + 6 K)}}{\sqrt[4]{3}}$ .

$y = \pm \frac{\sqrt[4]{2 (18 x - x^{3} + 6 K)}}{\sqrt[4]{3}}$

Step 3.4.10

Multiply $\frac{\sqrt[4]{2 (18 x - x^{3} + 6 K)}}{\sqrt[4]{3}}$ by $\frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$ .

$y = \pm \frac{\sqrt[4]{2 (18 x - x^{3} + 6 K)}}{\sqrt[4]{3}} \cdot \frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$

Step 3.4.11

Combine and simplify the denominator.

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Step 3.4.11.1

Multiply $\frac{\sqrt[4]{2 (18 x - x^{3} + 6 K)}}{\sqrt[4]{3}}$ by $\frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$ .

$y = \pm \frac{\sqrt[4]{2 (18 x - x^{3} + 6 K)} {\sqrt[4]{3}}^{3}}{\sqrt[4]{3} {\sqrt[4]{3}}^{3}}$

Step 3.4.11.2

Raise $\sqrt[4]{3}$ to the power of $1$ .

$y = \pm \frac{\sqrt[4]{2 (18 x - x^{3} + 6 K)} {\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{1} {\sqrt[4]{3}}^{3}}$

Step 3.4.11.3

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

$y = \pm \frac{\sqrt[4]{2 (18 x - x^{3} + 6 K)} {\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{1 + 3}}$

Step 3.4.11.4

Add $1$ and $3$ .

$y = \pm \frac{\sqrt[4]{2 (18 x - x^{3} + 6 K)} {\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{4}}$

Step 3.4.11.5

Rewrite ${\sqrt[4]{3}}^{4}$ as $3$ .

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Step 3.4.11.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3}$ as $3^{\frac{1}{4}}$ .

$y = \pm \frac{\sqrt[4]{2 (18 x - x^{3} + 6 K)} {\sqrt[4]{3}}^{3}}{{(3^{\frac{1}{4}})}^{4}}$

Step 3.4.11.5.2

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

$y = \pm \frac{\sqrt[4]{2 (18 x - x^{3} + 6 K)} {\sqrt[4]{3}}^{3}}{3^{\frac{1}{4} \cdot 4}}$

Step 3.4.11.5.3

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

$y = \pm \frac{\sqrt[4]{2 (18 x - x^{3} + 6 K)} {\sqrt[4]{3}}^{3}}{3^{\frac{4}{4}}}$

Step 3.4.11.5.4

Cancel the common factor of $4$ .

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Step 3.4.11.5.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt[4]{2 (18 x - x^{3} + 6 K)} {\sqrt[4]{3}}^{3}}{3^{\frac{4}{4}}}$

Step 3.4.11.5.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt[4]{2 (18 x - x^{3} + 6 K)} {\sqrt[4]{3}}^{3}}{3^{1}}$

Step 3.4.11.5.5

Evaluate the exponent.

$y = \pm \frac{\sqrt[4]{2 (18 x - x^{3} + 6 K)} {\sqrt[4]{3}}^{3}}{3}$

Step 3.4.12

Simplify the numerator.

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Step 3.4.12.1

Rewrite ${\sqrt[4]{3}}^{3}$ as $\sqrt[4]{3^{3}}$ .

$y = \pm \frac{\sqrt[4]{2 (18 x - x^{3} + 6 K)} \sqrt[4]{3^{3}}}{3}$

Step 3.4.12.2

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

$y = \pm \frac{\sqrt[4]{2 (18 x - x^{3} + 6 K)} \sqrt[4]{27}}{3}$

Step 3.4.13

Simplify the numerator.

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Step 3.4.13.1

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt[4]{2 (18 x - x^{3} + 6 K) \cdot 27}}{3}$

Step 3.4.13.2

Multiply $27$ by $2$ .

$y = \pm \frac{\sqrt[4]{54 (18 x - x^{3} + 6 K)}}{3}$

Step 3.5

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

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Step 3.5.1

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

$y = \frac{\sqrt[4]{54 (18 x - x^{3} + 6 K)}}{3}$

Step 3.5.2

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

$y = - \frac{\sqrt[4]{54 (18 x - x^{3} + 6 K)}}{3}$

Step 3.5.3

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

$y = \frac{\sqrt[4]{54 (18 x - x^{3} + 6 K)}}{3}$

$y = - \frac{\sqrt[4]{54 (18 x - x^{3} + 6 K)}}{3}$

$y = \frac{\sqrt[4]{54 (18 x - x^{3} + 6 K)}}{3}$

$y = - \frac{\sqrt[4]{54 (18 x - x^{3} + 6 K)}}{3}$

$y = \frac{\sqrt[4]{54 (18 x - x^{3} + 6 K)}}{3}$

$y = - \frac{\sqrt[4]{54 (18 x - x^{3} + 6 K)}}{3}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt[4]{54 (18 x - x^{3} + K)}}{3}$

$y = - \frac{\sqrt[4]{54 (18 x - x^{3} + K)}}{3}$