Calculus Examples

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

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{4}{x}$ .

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Step 1.1

Set up the integration.

$e^{\int \frac{4}{x} d x}$

Step 1.2

Integrate $\frac{4}{x}$ .

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Step 1.2.1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$e^{4 \int \frac{1}{x} d x}$

Step 1.2.2

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

$e^{4 (\ln (| x |) + C)}$

Step 1.2.3

Simplify.

$e^{4 \ln (| x |) + C}$

Step 1.3

Remove the constant of integration.

$e^{4 \ln (x)}$

Step 1.4

Use the logarithmic power rule.

$e^{\ln (x^{4})}$

Step 1.5

Exponentiation and log are inverse functions.

$x^{4}$

Step 2

Multiply each term by the integrating factor $x^{4}$ .

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Step 2.1

Multiply each term by $x^{4}$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

Combine $\frac{4}{x}$ and $y$ .

$x^{4} \frac{d y}{d x} + x^{4} \frac{4 y}{x} = x^{4} x^{2} + x^{4} \cdot - 1$

Step 2.2.2

Cancel the common factor of $x$ .

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Step 2.2.2.1

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

$x^{4} \frac{d y}{d x} + x \cdot x^{3} \frac{4 y}{x} = x^{4} x^{2} + x^{4} \cdot - 1$

Step 2.2.2.2

Cancel the common factor.

$x^{4} \frac{d y}{d x} + x \cdot x^{3} \frac{4 y}{x} = x^{4} x^{2} + x^{4} \cdot - 1$

Step 2.2.2.3

Rewrite the expression.

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

Step 2.2.3

Rewrite using the commutative property of multiplication.

$x^{4} \frac{d y}{d x} + 4 x^{3} y = x^{4} x^{2} + x^{4} \cdot - 1$

Step 2.3

Simplify each term.

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Step 2.3.1

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 2.3.1.1

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

$x^{4} \frac{d y}{d x} + 4 x^{3} y = x^{4 + 2} + x^{4} \cdot - 1$

Step 2.3.1.2

Add $4$ and $2$ .

$x^{4} \frac{d y}{d x} + 4 x^{3} y = x^{6} + x^{4} \cdot - 1$

Step 2.3.2

Move $- 1$ to the left of $x^{4}$ .

$x^{4} \frac{d y}{d x} + 4 x^{3} y = x^{6} - 1 \cdot x^{4}$

Step 2.3.3

Rewrite $- 1 x^{4}$ as $- x^{4}$ .

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

Step 3

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

$\frac{d}{d x} [x^{4} y] = x^{6} - x^{4}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [x^{4} y] d x = \int x^{6} - x^{4} d x$

Step 5

Integrate the left side.

$x^{4} y = \int x^{6} - x^{4} d x$

Step 6

Integrate the right side.

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Step 6.1

Split the single integral into multiple integrals.

$x^{4} y = \int x^{6} d x + \int - x^{4} d x$

Step 6.2

By the Power Rule, the integral of $x^{6}$ with respect to $x$ is $\frac{1}{7} x^{7}$ .

$x^{4} y = \frac{1}{7} x^{7} + C + \int - x^{4} d x$

Step 6.3

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

$x^{4} y = \frac{1}{7} x^{7} + C - \int x^{4} d x$

Step 6.4

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

$x^{4} y = \frac{1}{7} x^{7} + C - (\frac{1}{5} x^{5} + C)$

Step 6.5

Simplify.

$x^{4} y = \frac{1}{7} x^{7} - \frac{1}{5} x^{5} + C$

Step 7

Divide each term in $x^{4} y = \frac{1}{7} x^{7} - \frac{1}{5} x^{5} + C$ by $x^{4}$ and simplify.

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Step 7.1

Divide each term in $x^{4} y = \frac{1}{7} x^{7} - \frac{1}{5} x^{5} + C$ by $x^{4}$ .

$\frac{x^{4} y}{x^{4}} = \frac{\frac{1}{7} x^{7}}{x^{4}} + \frac{- \frac{1}{5} x^{5}}{x^{4}} + \frac{C}{x^{4}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

$\frac{x^{4} y}{x^{4}} = \frac{\frac{1}{7} x^{7}}{x^{4}} + \frac{- \frac{1}{5} x^{5}}{x^{4}} + \frac{C}{x^{4}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{7} x^{7}}{x^{4}} + \frac{- \frac{1}{5} x^{5}}{x^{4}} + \frac{C}{x^{4}}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Cancel the common factor of $x^{7}$ and $x^{4}$ .

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Step 7.3.1.1.1

Factor $x^{4}$ out of $\frac{1}{7} x^{7}$ .

$y = \frac{x^{4} (\frac{1}{7} x^{3})}{x^{4}} + \frac{- \frac{1}{5} x^{5}}{x^{4}} + \frac{C}{x^{4}}$

Step 7.3.1.1.2

Cancel the common factors.

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Step 7.3.1.1.2.1

Multiply by $1$ .

$y = \frac{x^{4} (\frac{1}{7} x^{3})}{x^{4} \cdot 1} + \frac{- \frac{1}{5} x^{5}}{x^{4}} + \frac{C}{x^{4}}$

Step 7.3.1.1.2.2

Cancel the common factor.

$y = \frac{x^{4} (\frac{1}{7} x^{3})}{x^{4} \cdot 1} + \frac{- \frac{1}{5} x^{5}}{x^{4}} + \frac{C}{x^{4}}$

Step 7.3.1.1.2.3

Rewrite the expression.

$y = \frac{\frac{1}{7} x^{3}}{1} + \frac{- \frac{1}{5} x^{5}}{x^{4}} + \frac{C}{x^{4}}$

Step 7.3.1.1.2.4

Divide $\frac{1}{7} x^{3}$ by $1$ .

$y = \frac{1}{7} x^{3} + \frac{- \frac{1}{5} x^{5}}{x^{4}} + \frac{C}{x^{4}}$

Step 7.3.1.2

Combine $\frac{1}{7}$ and $x^{3}$ .

$y = \frac{x^{3}}{7} + \frac{- \frac{1}{5} x^{5}}{x^{4}} + \frac{C}{x^{4}}$

Step 7.3.1.3

Cancel the common factor of $x^{5}$ and $x^{4}$ .

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Step 7.3.1.3.1

Factor $x^{4}$ out of $- \frac{1}{5} x^{5}$ .

$y = \frac{x^{3}}{7} + \frac{x^{4} (- \frac{1}{5} x)}{x^{4}} + \frac{C}{x^{4}}$

Step 7.3.1.3.2

Cancel the common factors.

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Step 7.3.1.3.2.1

Multiply by $1$ .

$y = \frac{x^{3}}{7} + \frac{x^{4} (- \frac{1}{5} x)}{x^{4} \cdot 1} + \frac{C}{x^{4}}$

Step 7.3.1.3.2.2

Cancel the common factor.

$y = \frac{x^{3}}{7} + \frac{x^{4} (- \frac{1}{5} x)}{x^{4} \cdot 1} + \frac{C}{x^{4}}$

Step 7.3.1.3.2.3

Rewrite the expression.

$y = \frac{x^{3}}{7} + \frac{- \frac{1}{5} x}{1} + \frac{C}{x^{4}}$

Step 7.3.1.3.2.4

Divide $- \frac{1}{5} x$ by $1$ .

$y = \frac{x^{3}}{7} - \frac{1}{5} x + \frac{C}{x^{4}}$

Step 7.3.1.4

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

$y = \frac{x^{3}}{7} - \frac{x}{5} + \frac{C}{x^{4}}$