Calculus Examples

$(10 + x^{4}) \frac{d y}{d x} = \frac{x^{3}}{y}$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $(10 + x^{4}) \frac{d y}{d x} = \frac{x^{3}}{y}$ by $10 + x^{4}$ and simplify.

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Step 1.1.1

Divide each term in $(10 + x^{4}) \frac{d y}{d x} = \frac{x^{3}}{y}$ by $10 + x^{4}$ .

$\frac{(10 + x^{4}) \frac{d y}{d x}}{10 + x^{4}} = \frac{\frac{x^{3}}{y}}{10 + x^{4}}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $10 + x^{4}$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{(10 + x^{4}) \frac{d y}{d x}}{10 + x^{4}} = \frac{\frac{x^{3}}{y}}{10 + x^{4}}$

Step 1.1.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} = \frac{\frac{x^{3}}{y}}{10 + x^{4}}$

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{d y}{d x} = \frac{x^{3}}{y} \cdot \frac{1}{10 + x^{4}}$

Step 1.1.3.2

Multiply $\frac{x^{3}}{y}$ by $\frac{1}{10 + x^{4}}$ .

$\frac{d y}{d x} = \frac{x^{3}}{y (10 + x^{4})}$

Step 1.2

Regroup factors.

$\frac{d y}{d x} = \frac{x^{3}}{10 + x^{4}} \cdot \frac{1}{y}$

Step 1.3

Multiply both sides by $y$ .

$y \frac{d y}{d x} = y (\frac{x^{3}}{10 + x^{4}} \cdot \frac{1}{y})$

Step 1.4

Simplify.

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Step 1.4.1

Multiply $\frac{x^{3}}{10 + x^{4}}$ by $\frac{1}{y}$ .

$y \frac{d y}{d x} = y \frac{x^{3}}{(10 + x^{4}) y}$

Step 1.4.2

Cancel the common factor of $y$ .

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Step 1.4.2.1

Factor $y$ out of $(10 + x^{4}) y$ .

$y \frac{d y}{d x} = y \frac{x^{3}}{y (10 + x^{4})}$

Step 1.4.2.2

Cancel the common factor.

$y \frac{d y}{d x} = y \frac{x^{3}}{y (10 + x^{4})}$

Step 1.4.2.3

Rewrite the expression.

$y \frac{d y}{d x} = \frac{x^{3}}{10 + x^{4}}$

Step 1.5

Rewrite the equation.

$y d y = \frac{x^{3}}{10 + x^{4}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int \frac{x^{3}}{10 + x^{4}} d x$

Step 2.2

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

$\frac{1}{2} y^{2} + C_{1} = \int \frac{x^{3}}{10 + x^{4}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Rewrite $x^{4}$ as ${(x^{4})}^{1}$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{x^{3}}{10 + {(x^{4})}^{1}} d x$

Step 2.3.2

Let $u_{1} = x^{4}$ . Then $d u_{1} = 4 x^{3} d x$ , so $\frac{1}{4} d u_{1} = x^{3} d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.3.2.1

Let $u_{1} = x^{4}$ . Find $\frac{d u_{1}}{d x}$ .

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Step 2.3.2.1.1

Differentiate $x^{4}$ .

$\frac{d}{d x} [x^{4}]$

Step 2.3.2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$4 x^{3}$

Step 2.3.2.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{1}{10 + {u_{1}}^{1}} \cdot \frac{1}{4} d u_{1}$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Simplify.

$\frac{1}{2} y^{2} + C_{1} = \int \frac{1}{10 + u_{1}} \cdot \frac{1}{4} d u_{1}$

Step 2.3.3.2

Multiply $\frac{1}{10 + u_{1}}$ by $\frac{1}{4}$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{1}{(10 + u_{1}) \cdot 4} d u_{1}$

Step 2.3.3.3

Move $4$ to the left of $10 + u_{1}$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{1}{4 (10 + u_{1})} d u_{1}$

Step 2.3.4

Since $\frac{1}{4}$ is constant with respect to $u_{1}$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{4} \int \frac{1}{10 + u_{1}} d u_{1}$

Step 2.3.5

Let $u_{2} = 10 + u_{1}$ . Then $d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.3.5.1

Let $u_{2} = 10 + u_{1}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 2.3.5.1.1

Differentiate $10 + u_{1}$ .

$\frac{d}{d u_{1}} [10 + u_{1}]$

Step 2.3.5.1.2

By the Sum Rule, the derivative of $10 + u_{1}$ with respect to $u_{1}$ is $\frac{d}{d u_{1}} [10] + \frac{d}{d u_{1}} [u_{1}]$ .

$\frac{d}{d u_{1}} [10] + \frac{d}{d u_{1}} [u_{1}]$

Step 2.3.5.1.3

Since $10$ is constant with respect to $u_{1}$ , the derivative of $10$ with respect to $u_{1}$ is $0$ .

$0 + \frac{d}{d u_{1}} [u_{1}]$

Step 2.3.5.1.4

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

$0 + 1$

Step 2.3.5.1.5

Add $0$ and $1$ .

$1$

Step 2.3.5.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{4} \int \frac{1}{u_{2}} d u_{2}$

Step 2.3.6

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

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

Step 2.3.7

Simplify.

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

Step 2.3.8

Substitute back in for each integration substitution variable.

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Step 2.3.8.1

Replace all occurrences of $u_{2}$ with $10 + u_{1}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{4} \ln (| 10 + u_{1} |) + C_{2}$

Step 2.3.8.2

Replace all occurrences of $u_{1}$ with $x^{4}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{4} \ln (| 10 + x^{4} |) + C_{2}$

Step 2.4

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

$\frac{1}{2} y^{2} = \frac{1}{4} \ln (| 10 + x^{4} |) + C$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (\frac{1}{4} \ln (| 10 + x^{4} |) + C)$

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 $2 (\frac{1}{2} y^{2})$ .

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Step 3.2.1.1.1

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

$2 \frac{y^{2}}{2} = 2 (\frac{1}{4} \ln (| 10 + x^{4} |) + C)$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$2 \frac{y^{2}}{2} = 2 (\frac{1}{4} \ln (| 10 + x^{4} |) + C)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (\frac{1}{4} \ln (| 10 + x^{4} |) + C)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (\frac{1}{4} \ln (| 10 + x^{4} |) + C)$ .

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Step 3.2.2.1.1

Combine $\frac{1}{4}$ and $\ln (| 10 + x^{4} |)$ .

$y^{2} = 2 (\frac{\ln (| 10 + x^{4} |)}{4} + C)$

Step 3.2.2.1.2

To write $C$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y^{2} = 2 (\frac{\ln (| 10 + x^{4} |)}{4} + C \cdot \frac{4}{4})$

Step 3.2.2.1.3

Simplify terms.

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Step 3.2.2.1.3.1

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

$y^{2} = 2 (\frac{\ln (| 10 + x^{4} |)}{4} + \frac{C \cdot 4}{4})$

Step 3.2.2.1.3.2

Combine the numerators over the common denominator.

$y^{2} = 2 \frac{\ln (| 10 + x^{4} |) + C \cdot 4}{4}$

Step 3.2.2.1.3.3

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.3.1

Factor $2$ out of $4$ .

$y^{2} = 2 \frac{\ln (| 10 + x^{4} |) + C \cdot 4}{2 (2)}$

Step 3.2.2.1.3.3.2

Cancel the common factor.

$y^{2} = 2 \frac{\ln (| 10 + x^{4} |) + C \cdot 4}{2 \cdot 2}$

Step 3.2.2.1.3.3.3

Rewrite the expression.

$y^{2} = \frac{\ln (| 10 + x^{4} |) + C \cdot 4}{2}$

Step 3.2.2.1.4

Move $4$ to the left of $C$ .

$y^{2} = \frac{\ln (| 10 + x^{4} |) + 4 C}{2}$

Step 3.3

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

$y = \pm \sqrt{\frac{\ln (| 10 + x^{4} |) + 4 C}{2}}$

Step 3.4

Simplify $\pm \sqrt{\frac{\ln (| 10 + x^{4} |) + 4 C}{2}}$ .

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Step 3.4.1

Rewrite $\sqrt{\frac{\ln (| 10 + x^{4} |) + 4 C}{2}}$ as $\frac{\sqrt{\ln (| 10 + x^{4} |) + 4 C}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{\ln (| 10 + x^{4} |) + 4 C}}{\sqrt{2}}$

Step 3.4.2

Multiply $\frac{\sqrt{\ln (| 10 + x^{4} |) + 4 C}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{\ln (| 10 + x^{4} |) + 4 C}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 3.4.3

Combine and simplify the denominator.

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Step 3.4.3.1

Multiply $\frac{\sqrt{\ln (| 10 + x^{4} |) + 4 C}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{\ln (| 10 + x^{4} |) + 4 C} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 3.4.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$y = \pm \frac{\sqrt{\ln (| 10 + x^{4} |) + 4 C} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 3.4.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$y = \pm \frac{\sqrt{\ln (| 10 + x^{4} |) + 4 C} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 3.4.3.4

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

$y = \pm \frac{\sqrt{\ln (| 10 + x^{4} |) + 4 C} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 3.4.3.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{\ln (| 10 + x^{4} |) + 4 C} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 3.4.3.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{\ln (| 10 + x^{4} |) + 4 C} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 3.4.3.6.2

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

$y = \pm \frac{\sqrt{\ln (| 10 + x^{4} |) + 4 C} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 3.4.3.6.3

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

$y = \pm \frac{\sqrt{\ln (| 10 + x^{4} |) + 4 C} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.4.3.6.4

Cancel the common factor of $2$ .

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Step 3.4.3.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{\ln (| 10 + x^{4} |) + 4 C} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.4.3.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{\ln (| 10 + x^{4} |) + 4 C} \sqrt{2}}{2^{1}}$

Step 3.4.3.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{\ln (| 10 + x^{4} |) + 4 C} \sqrt{2}}{2}$

Step 3.4.4

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(\ln (| 10 + x^{4} |) + 4 C) \cdot 2}}{2}$

Step 3.4.5

Reorder factors in $\pm \frac{\sqrt{(\ln (| 10 + x^{4} |) + 4 C) \cdot 2}}{2}$ .

$y = \pm \frac{\sqrt{2 (\ln (| 10 + x^{4} |) + 4 C)}}{2}$

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.

$y = \frac{\sqrt{2 (\ln (| 10 + x^{4} |) + 4 C)}}{2}$

Step 3.5.2

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

$y = - \frac{\sqrt{2 (\ln (| 10 + x^{4} |) + 4 C)}}{2}$

Step 3.5.3

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

$y = \frac{\sqrt{2 (\ln (| 10 + x^{4} |) + 4 C)}}{2}$

$y = - \frac{\sqrt{2 (\ln (| 10 + x^{4} |) + 4 C)}}{2}$

$y = \frac{\sqrt{2 (\ln (| 10 + x^{4} |) + 4 C)}}{2}$

$y = - \frac{\sqrt{2 (\ln (| 10 + x^{4} |) + 4 C)}}{2}$

$y = \frac{\sqrt{2 (\ln (| 10 + x^{4} |) + 4 C)}}{2}$

$y = - \frac{\sqrt{2 (\ln (| 10 + x^{4} |) + 4 C)}}{2}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt{2 (\ln (| 10 + x^{4} |) + C)}}{2}$

$y = - \frac{\sqrt{2 (\ln (| 10 + x^{4} |) + C)}}{2}$