Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

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

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

Step 1.3

Factor $y$ out of $\frac{(3 x + 1) y}{x}$ .

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

Step 1.4

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

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

Step 2

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

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Step 2.1

Set up the integration.

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

Step 2.2

Integrate $\frac{3 x + 1}{x}$ .

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Step 2.2.1

Split the fraction into multiple fractions.

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

Step 2.2.2

Split the single integral into multiple integrals.

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

Step 2.2.3

Cancel the common factor of $x$ .

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Step 2.2.3.1

Cancel the common factor.

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

Step 2.2.3.2

Divide $3$ by $1$ .

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

Step 2.2.4

Apply the constant rule.

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

Step 2.2.5

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

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

Step 2.2.6

Simplify.

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

Step 2.3

Remove the constant of integration.

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

Step 3

Multiply each term by the integrating factor $e^{3 x + \ln (x)}$ .

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Step 3.1

Multiply each term by $e^{3 x + \ln (x)}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Combine $e^{3 x + \ln (x)}$ and $\frac{(3 x + 1) y}{x}$ .

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

Step 3.3

To write $e^{3 x + \ln (x)} \frac{d y}{d x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

Factor $e^{3 x + \ln (x)}$ out of $e^{3 x + \ln (x)} \frac{d y}{d x} x + e^{3 x + \ln (x)} (3 x + 1) y$ .

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Step 3.5.1.1

Factor $e^{3 x + \ln (x)}$ out of $e^{3 x + \ln (x)} \frac{d y}{d x} x$ .

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

Step 3.5.1.2

Factor $e^{3 x + \ln (x)}$ out of $e^{3 x + \ln (x)} (3 x + 1) y$ .

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

Step 3.5.1.3

Factor $e^{3 x + \ln (x)}$ out of $e^{3 x + \ln (x)} (\frac{d y}{d x} x) + e^{3 x + \ln (x)} ((3 x + 1) y)$ .

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

Step 3.5.2

Apply the distributive property.

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

Step 3.5.3

Multiply $y$ by $1$ .

$\frac{e^{3 x + \ln (x)} (\frac{d y}{d x} x + 3 x y + y)}{x} = e^{3 x + \ln (x)} \frac{e^{- 3 x}}{x}$

Step 3.6

Multiply $e^{3 x + \ln (x)} \frac{e^{- 3 x}}{x}$ .

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Step 3.6.1

Combine $e^{3 x + \ln (x)}$ and $\frac{e^{- 3 x}}{x}$ .

$\frac{e^{3 x + \ln (x)} (\frac{d y}{d x} x + 3 x y + y)}{x} = \frac{e^{3 x + \ln (x)} e^{- 3 x}}{x}$

Step 3.6.2

Multiply $e^{3 x + \ln (x)}$ by $e^{- 3 x}$ by adding the exponents.

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Step 3.6.2.1

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

$\frac{e^{3 x + \ln (x)} (\frac{d y}{d x} x + 3 x y + y)}{x} = \frac{e^{3 x + \ln (x) - 3 x}}{x}$

Step 3.6.2.2

Combine the opposite terms in $3 x + \ln (x) - 3 x$ .

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Step 3.6.2.2.1

Subtract $3 x$ from $3 x$ .

$\frac{e^{3 x + \ln (x)} (\frac{d y}{d x} x + 3 x y + y)}{x} = \frac{e^{0 + \ln (x)}}{x}$

Step 3.6.2.2.2

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

$\frac{e^{3 x + \ln (x)} (\frac{d y}{d x} x + 3 x y + y)}{x} = \frac{e^{\ln (x)}}{x}$

Step 3.7

Exponentiation and log are inverse functions.

$\frac{e^{3 x + \ln (x)} (\frac{d y}{d x} x + 3 x y + y)}{x} = \frac{x}{x}$

Step 3.8

Cancel the common factor of $x$ .

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Step 3.8.1

Cancel the common factor.

$\frac{e^{3 x + \ln (x)} (\frac{d y}{d x} x + 3 x y + y)}{x} = \frac{x}{x}$

Step 3.8.2

Rewrite the expression.

$\frac{e^{3 x + \ln (x)} (\frac{d y}{d x} x + 3 x y + y)}{x} = 1$

Step 3.9

Reorder factors in $\frac{e^{3 x + \ln (x)} (\frac{d y}{d x} x + 3 x y + y)}{x} = 1$ .

$\frac{e^{3 x + \ln (x)} (x \frac{d y}{d x} + 3 x y + y)}{x} = 1$

Step 4

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

$\frac{d}{d x} [e^{3 x + \ln (x)} y] = 1$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [e^{3 x + \ln (x)} y] d x = \int d x$

Step 6

Integrate the left side.

$e^{3 x + \ln (x)} y = \int d x$

Step 7

Apply the constant rule.

$e^{3 x + \ln (x)} y = x + C$

Step 8

Divide each term in $e^{3 x + \ln (x)} y = x + C$ by $e^{3 x + \ln (x)}$ and simplify.

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Step 8.1

Divide each term in $e^{3 x + \ln (x)} y = x + C$ by $e^{3 x + \ln (x)}$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $e^{3 x + \ln (x)}$ .

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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