Calculus Examples

$\frac{x^{5} + 3 y}{x} - \frac{d y}{d x} = 0$

Step 1

Rewrite the differential equation as $\frac{d y}{d x} + M (x) y = Q (x)$ .

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Step 1.1

Rewrite the equation as $M (x) \frac{d y}{d x} + P (x) y = Q (x)$ .

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Step 1.1.1

Split the fraction $\frac{x^{5} + 3 y}{x}$ into two fractions.

$\frac{x^{5}}{x} + \frac{3 y}{x} - \frac{d y}{d x} = 0$

Step 1.1.2

Subtract $\frac{x^{5}}{x}$ from both sides of the equation.

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

Step 1.1.3

Reorder terms.

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

Step 1.2

Factor $y$ out of $\frac{3 y}{x}$ .

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

Step 1.3

Reorder $y$ and $\frac{3}{x}$ .

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

Step 1.4

Divide each term in $- \frac{d y}{d x} + \frac{3}{x} y = - \frac{x^{5}}{x}$ by $- 1$ .

$\frac{- \frac{d y}{d x}}{- 1} + \frac{\frac{3}{x} y}{- 1} = \frac{- \frac{x^{5}}{x}}{- 1}$

Step 1.5

Dividing two negative values results in a positive value.

$\frac{\frac{d y}{d x}}{1} + \frac{\frac{3}{x} y}{- 1} = \frac{- \frac{x^{5}}{x}}{- 1}$

Step 1.6

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

$\frac{d y}{d x} + \frac{\frac{3}{x} y}{- 1} = \frac{- \frac{x^{5}}{x}}{- 1}$

Step 1.7

Move the negative one from the denominator of $\frac{\frac{3}{x} y}{- 1}$ .

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

Step 1.8

Cancel the common factor of $x^{5}$ and $x$ .

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Step 1.8.1

Factor $x$ out of $x^{5}$ .

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

Step 1.8.2

Cancel the common factors.

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Step 1.8.2.1

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

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

Step 1.8.2.2

Factor $x$ out of $x^{1}$ .

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

Step 1.8.2.3

Cancel the common factor.

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

Step 1.8.2.4

Rewrite the expression.

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

Step 1.8.2.5

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

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

Step 1.9

Dividing two negative values results in a positive value.

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

Step 1.10

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

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

Step 1.11

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

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

Step 1.12

Factor $y$ out of $- 1 \cdot \frac{3 y}{x}$ .

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

Step 1.13

Reorder $y$ and $- 1 \cdot \frac{3}{x}$ .

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = - 1 \cdot \frac{3}{x}$ .

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Step 2.1

Set up the integration.

$e^{\int - 1 \cdot \frac{3}{x} d x}$

Step 2.2

Integrate $- 1 \cdot \frac{3}{x}$ .

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Step 2.2.1

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

$e^{- \int \frac{3}{x} d x}$

Step 2.2.2

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

$e^{- (3 \int \frac{1}{x} d x)}$

Step 2.2.3

Multiply $3$ by $- 1$ .

$e^{- 3 \int \frac{1}{x} d x}$

Step 2.2.4

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

$e^{- 3 (\ln (| x |) + C)}$

Step 2.2.5

Simplify.

$e^{- 3 \ln (| x |) + C}$

Step 2.3

Remove the constant of integration.

$e^{- 3 \ln (x)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{- 3})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{- 3}$

Step 2.6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{3}}$

Step 3

Multiply each term by the integrating factor $\frac{1}{x^{3}}$ .

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Step 3.1

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

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Rewrite using the commutative property of multiplication.

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

Step 3.2.3

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

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

Step 3.2.4

Multiply $- \frac{1}{x^{3}} \cdot \frac{3 y}{x}$ .

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Step 3.2.4.1

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

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

Step 3.2.4.2

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 3.2.4.2.1

Multiply $x$ by $x^{3}$ .

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Step 3.2.4.2.1.1

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

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

Step 3.2.4.2.1.2

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

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

Step 3.2.4.2.2

Add $1$ and $3$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 4

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [\frac{1}{x^{3}} y] = x$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\frac{1}{x^{3}} y] d x = \int x d x$

Step 6

Integrate the left side.

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

Step 7

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

$\frac{1}{x^{3}} y = \frac{1}{2} x^{2} + C$

Step 8

Solve for $y$ .

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Step 8.1

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

$\frac{y}{x^{3}} = \frac{1}{2} x^{2} + C$

Step 8.2

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

$\frac{y}{x^{3}} = \frac{x^{2}}{2} + C$

Step 8.3

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

$\frac{y}{x^{3}} x^{3} = (\frac{x^{2}}{2} + C) x^{3}$

Step 8.4

Simplify.

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Step 8.4.1

Simplify the left side.

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Step 8.4.1.1

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

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Step 8.4.1.1.1

Cancel the common factor.

$\frac{y}{x^{3}} x^{3} = (\frac{x^{2}}{2} + C) x^{3}$

Step 8.4.1.1.2

Rewrite the expression.

$y = (\frac{x^{2}}{2} + C) x^{3}$

Step 8.4.2

Simplify the right side.

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Step 8.4.2.1

Simplify $(\frac{x^{2}}{2} + C) x^{3}$ .

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Step 8.4.2.1.1

Apply the distributive property.

$y = \frac{x^{2}}{2} x^{3} + C x^{3}$

Step 8.4.2.1.2

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

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Step 8.4.2.1.2.1

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

$y = \frac{x^{2} x^{3}}{2} + C x^{3}$

Step 8.4.2.1.2.2

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 8.4.2.1.2.2.1

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

$y = \frac{x^{2 + 3}}{2} + C x^{3}$

Step 8.4.2.1.2.2.2

Add $2$ and $3$ .

$y = \frac{x^{5}}{2} + C x^{3}$