Calculus Examples

$y^{5} \frac{d y}{d x} = 3 x^{4} y^{6} - 6 x^{4}$

Step 1

Separate the variables.

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Step 1.1

Factor $3 x^{4}$ out of $3 x^{4} y^{6} - 6 x^{4}$ .

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Step 1.1.1

Factor $3 x^{4}$ out of $3 x^{4} y^{6}$ .

$y^{5} \frac{d y}{d x} = 3 x^{4} (y^{6}) - 6 x^{4}$

Step 1.1.2

Factor $3 x^{4}$ out of $- 6 x^{4}$ .

$y^{5} \frac{d y}{d x} = 3 x^{4} (y^{6}) + 3 x^{4} (- 2)$

Step 1.1.3

Factor $3 x^{4}$ out of $3 x^{4} (y^{6}) + 3 x^{4} (- 2)$ .

$y^{5} \frac{d y}{d x} = 3 x^{4} (y^{6} - 2)$

Step 1.2

Multiply both sides by $\frac{1}{y^{6} - 2}$ .

$\frac{1}{y^{6} - 2} (y^{5} \frac{d y}{d x}) = \frac{1}{y^{6} - 2} (3 x^{4} (y^{6} - 2))$

Step 1.3

Simplify.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

$\frac{1}{y^{6} - 2} (y^{5} \frac{d y}{d x}) = 3 \frac{1}{y^{6} - 2} (x^{4} (y^{6} - 2))$

Step 1.3.2

Combine $3$ and $\frac{1}{y^{6} - 2}$ .

$\frac{1}{y^{6} - 2} (y^{5} \frac{d y}{d x}) = \frac{3}{y^{6} - 2} (x^{4} (y^{6} - 2))$

Step 1.3.3

Cancel the common factor of $y^{6} - 2$ .

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Step 1.3.3.1

Factor $y^{6} - 2$ out of $x^{4} (y^{6} - 2)$ .

$\frac{1}{y^{6} - 2} (y^{5} \frac{d y}{d x}) = \frac{3}{y^{6} - 2} ((y^{6} - 2) x^{4})$

Step 1.3.3.2

Cancel the common factor.

$\frac{1}{y^{6} - 2} (y^{5} \frac{d y}{d x}) = \frac{3}{y^{6} - 2} ((y^{6} - 2) x^{4})$

Step 1.3.3.3

Rewrite the expression.

$\frac{1}{y^{6} - 2} (y^{5} \frac{d y}{d x}) = 3 x^{4}$

Step 1.4

Remove unnecessary parentheses.

$\frac{1}{y^{6} - 2} y^{5} \frac{d y}{d x} = 3 x^{4}$

Step 1.5

Rewrite the equation.

$\frac{1}{y^{6} - 2} y^{5} d y = 3 x^{4} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y^{6} - 2} y^{5} d y = \int 3 x^{4} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Combine $\frac{1}{y^{6} - 2}$ and $y^{5}$ .

$\int \frac{y^{5}}{y^{6} - 2} d y = \int 3 x^{4} d x$

Step 2.2.2

Let $u = y^{6} - 2$ . Then $d u = 6 y^{5} d y$ , so $\frac{1}{6} d u = y^{5} d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.2.1

Let $u = y^{6} - 2$ . Find $\frac{d u}{d y}$ .

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Step 2.2.2.1.1

Differentiate $y^{6} - 2$ .

$\frac{d}{d y} [y^{6} - 2]$

Step 2.2.2.1.2

By the Sum Rule, the derivative of $y^{6} - 2$ with respect to $y$ is $\frac{d}{d y} [y^{6}] + \frac{d}{d y} [- 2]$ .

$\frac{d}{d y} [y^{6}] + \frac{d}{d y} [- 2]$

Step 2.2.2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 6$ .

$6 y^{5} + \frac{d}{d y} [- 2]$

Step 2.2.2.1.4

Since $- 2$ is constant with respect to $y$ , the derivative of $- 2$ with respect to $y$ is $0$ .

$6 y^{5} + 0$

Step 2.2.2.1.5

Add $6 y^{5}$ and $0$ .

$6 y^{5}$

Step 2.2.2.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{1}{u} \cdot \frac{1}{6} d u = \int 3 x^{4} d x$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Multiply $\frac{1}{u}$ by $\frac{1}{6}$ .

$\int \frac{1}{u \cdot 6} d u = \int 3 x^{4} d x$

Step 2.2.3.2

Move $6$ to the left of $u$ .

$\int \frac{1}{6 u} d u = \int 3 x^{4} d x$

Step 2.2.4

Since $\frac{1}{6}$ is constant with respect to $u$ , move $\frac{1}{6}$ out of the integral.

$\frac{1}{6} \int \frac{1}{u} d u = \int 3 x^{4} d x$

Step 2.2.5

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

$\frac{1}{6} (\ln (| u |) + C_{1}) = \int 3 x^{4} d x$

Step 2.2.6

Simplify.

$\frac{1}{6} \ln (| u |) + C_{1} = \int 3 x^{4} d x$

Step 2.2.7

Replace all occurrences of $u$ with $y^{6} - 2$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$\frac{1}{6} \ln (| y^{6} - 2 |) + C_{1} = 3 (\frac{1}{5} x^{5} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $3 (\frac{1}{5} x^{5} + C_{2})$ as $3 (\frac{1}{5}) x^{5} + C_{2}$ .

$\frac{1}{6} \ln (| y^{6} - 2 |) + C_{1} = 3 (\frac{1}{5}) x^{5} + C_{2}$

Step 2.3.3.2

Combine $3$ and $\frac{1}{5}$ .

$\frac{1}{6} \ln (| y^{6} - 2 |) + C_{1} = \frac{3}{5} x^{5} + C_{2}$

Step 2.4

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

$\frac{1}{6} \ln (| y^{6} - 2 |) = \frac{3}{5} x^{5} + C$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $6$ .

$6 (\frac{1}{6} \ln (| y^{6} - 2 |)) = 6 (\frac{3}{5} x^{5} + C)$

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 $6 (\frac{1}{6} \ln (| y^{6} - 2 |))$ .

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Step 3.2.1.1.1

Combine $\frac{1}{6}$ and $\ln (| y^{6} - 2 |)$ .

$6 \frac{\ln (| y^{6} - 2 |)}{6} = 6 (\frac{3}{5} x^{5} + C)$

Step 3.2.1.1.2

Cancel the common factor of $6$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$6 \frac{\ln (| y^{6} - 2 |)}{6} = 6 (\frac{3}{5} x^{5} + C)$

Step 3.2.1.1.2.2

Rewrite the expression.

$\ln (| y^{6} - 2 |) = 6 (\frac{3}{5} x^{5} + C)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $6 (\frac{3}{5} x^{5} + C)$ .

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Step 3.2.2.1.1

Combine $\frac{3}{5}$ and $x^{5}$ .

$\ln (| y^{6} - 2 |) = 6 (\frac{3 x^{5}}{5} + C)$

Step 3.2.2.1.2

Apply the distributive property.

$\ln (| y^{6} - 2 |) = 6 \frac{3 x^{5}}{5} + 6 C$

Step 3.2.2.1.3

Multiply $6 \frac{3 x^{5}}{5}$ .

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Step 3.2.2.1.3.1

Combine $6$ and $\frac{3 x^{5}}{5}$ .

$\ln (| y^{6} - 2 |) = \frac{6 (3 x^{5})}{5} + 6 C$

Step 3.2.2.1.3.2

Multiply $3$ by $6$ .

$\ln (| y^{6} - 2 |) = \frac{18 x^{5}}{5} + 6 C$

Step 3.3

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (| y^{6} - 2 |)} = e^{\frac{18 x^{5}}{5} + 6 C}$

Step 3.4

Rewrite $\ln (| y^{6} - 2 |) = \frac{18 x^{5}}{5} + 6 C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{\frac{18 x^{5}}{5} + 6 C} = | y^{6} - 2 |$

Step 3.5

Solve for $y$ .

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Step 3.5.1

Rewrite the equation as $| y^{6} - 2 | = e^{\frac{18 x^{5}}{5} + 6 C}$ .

$| y^{6} - 2 | = e^{\frac{18 x^{5}}{5} + 6 C}$

Step 3.5.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$y^{6} - 2 = \pm e^{\frac{18 x^{5}}{5} + 6 C}$

Step 3.5.3

Add $2$ to both sides of the equation.

$y^{6} = \pm e^{\frac{18 x^{5}}{5} + 6 C} + 2$

Step 3.5.4

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

$y = \pm \sqrt[6]{\pm e^{\frac{18 x^{5}}{5} + 6 C} + 2}$

Step 4

Group the constant terms together.

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Step 4.1

Simplify the constant of integration.

$y = \pm \sqrt[6]{\pm e^{\frac{18 x^{5}}{5} + C} + 2}$

Step 4.2

Rewrite $e^{\frac{18 x^{5}}{5} + C}$ as $e^{\frac{18 x^{5}}{5}} e^{C}$ .

$y = \pm \sqrt[6]{\pm e^{\frac{18 x^{5}}{5}} e^{C} + 2}$

Step 4.3

Reorder $e^{\frac{18 x^{5}}{5}}$ and $e^{C}$ .

$y = \pm \sqrt[6]{\pm e^{C} e^{\frac{18 x^{5}}{5}} + 2}$

Step 4.4

Combine constants with the plus or minus.

$y = \pm \sqrt[6]{C e^{\frac{18 x^{5}}{5}} + 2}$