Calculus Examples

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

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{2 x + 1}{x}$ .

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Step 1.1

Set up the integration.

$e^{\int \frac{2 x + 1}{x} d x}$

Step 1.2

Integrate $\frac{2 x + 1}{x}$ .

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Step 1.2.1

Split the fraction into multiple fractions.

$e^{\int \frac{2 x}{x} + \frac{1}{x} d x}$

Step 1.2.2

Split the single integral into multiple integrals.

$e^{\int \frac{2 x}{x} d x + \int \frac{1}{x} d x}$

Step 1.2.3

Cancel the common factor of $x$ .

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Step 1.2.3.1

Cancel the common factor.

$e^{\int \frac{2 x}{x} d x + \int \frac{1}{x} d x}$

Step 1.2.3.2

Divide $2$ by $1$ .

$e^{\int 2 d x + \int \frac{1}{x} d x}$

Step 1.2.4

Apply the constant rule.

$e^{2 x + C + \int \frac{1}{x} d x}$

Step 1.2.5

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

$e^{2 x + C + \ln (| x |) + C}$

Step 1.2.6

Simplify.

$e^{2 x + \ln (| x |) + C}$

Step 1.3

Remove the constant of integration.

$e^{2 x + \ln (x)}$

Step 2

Multiply each term by the integrating factor $e^{2 x + \ln (x)}$ .

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Step 2.1

Multiply each term by $e^{2 x + \ln (x)}$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

Combine $\frac{2 x + 1}{x}$ and $y$ .

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

Step 2.2.2

Combine $e^{2 x + \ln (x)}$ and $\frac{(2 x + 1) y}{x}$ .

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

Step 2.3

To write $e^{2 x + \ln (x)} \frac{d y}{d x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Factor $e^{2 x + \ln (x)}$ out of $e^{2 x + \ln (x)} \frac{d y}{d x} x + e^{2 x + \ln (x)} (2 x + 1) y$ .

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Step 2.5.1.1

Factor $e^{2 x + \ln (x)}$ out of $e^{2 x + \ln (x)} \frac{d y}{d x} x$ .

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

Step 2.5.1.2

Factor $e^{2 x + \ln (x)}$ out of $e^{2 x + \ln (x)} (2 x + 1) y$ .

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

Step 2.5.1.3

Factor $e^{2 x + \ln (x)}$ out of $e^{2 x + \ln (x)} (\frac{d y}{d x} x) + e^{2 x + \ln (x)} ((2 x + 1) y)$ .

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

Step 2.5.2

Apply the distributive property.

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

Step 2.5.3

Multiply $y$ by $1$ .

$\frac{e^{2 x + \ln (x)} (\frac{d y}{d x} x + 2 x y + y)}{x} = e^{2 x + \ln (x)} e^{- 2 x}$

Step 2.6

Multiply $e^{2 x + \ln (x)}$ by $e^{- 2 x}$ by adding the exponents.

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Step 2.6.1

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

$\frac{e^{2 x + \ln (x)} (\frac{d y}{d x} x + 2 x y + y)}{x} = e^{2 x + \ln (x) - 2 x}$

Step 2.6.2

Combine the opposite terms in $2 x + \ln (x) - 2 x$ .

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Step 2.6.2.1

Subtract $2 x$ from $2 x$ .

$\frac{e^{2 x + \ln (x)} (\frac{d y}{d x} x + 2 x y + y)}{x} = e^{0 + \ln (x)}$

Step 2.6.2.2

Add $0$ and $\ln (x)$ .

$\frac{e^{2 x + \ln (x)} (\frac{d y}{d x} x + 2 x y + y)}{x} = e^{\ln (x)}$

Step 2.7

Exponentiation and log are inverse functions.

$\frac{e^{2 x + \ln (x)} (\frac{d y}{d x} x + 2 x y + y)}{x} = x$

Step 2.8

Reorder factors in $\frac{e^{2 x + \ln (x)} (\frac{d y}{d x} x + 2 x y + y)}{x} = x$ .

$\frac{e^{2 x + \ln (x)} (x \frac{d y}{d x} + 2 x y + y)}{x} = x$

Step 3

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

$\frac{d}{d x} [e^{2 x + \ln (x)} y] = x$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{2 x + \ln (x)} y] d x = \int x d x$

Step 5

Integrate the left side.

$e^{2 x + \ln (x)} y = \int x d x$

Step 6

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

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

Step 7

Divide each term in $e^{2 x + \ln (x)} y = \frac{1}{2} x^{2} + C$ by $e^{2 x + \ln (x)}$ and simplify.

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Step 7.1

Divide each term in $e^{2 x + \ln (x)} y = \frac{1}{2} x^{2} + C$ by $e^{2 x + \ln (x)}$ .

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

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $e^{2 x + \ln (x)}$ .

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $y$ by $1$ .

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

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

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

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

Step 7.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{x^{2}}{2} \cdot \frac{1}{e^{2 x + \ln (x)}} + \frac{C}{e^{2 x + \ln (x)}}$

Step 7.3.1.3

Combine.

$y = \frac{x^{2} \cdot 1}{2 e^{2 x + \ln (x)}} + \frac{C}{e^{2 x + \ln (x)}}$

Step 7.3.1.4

Multiply $x^{2}$ by $1$ .

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