Calculus Examples

$\frac{d y}{d x} + (\sin (x)) y = e^{\cos (x)}$

Step 1

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

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Step 1.1

Set up the integration.

$e^{\int \sin (x) d x}$

Step 1.2

The integral of $\sin (x)$ with respect to $x$ is $- \cos (x)$ .

$e^{- \cos (x) + C}$

Step 1.3

Remove the constant of integration.

$e^{- \cos (x)}$

Step 2

Multiply each term by the integrating factor $e^{- \cos (x)}$ .

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Step 2.1

Multiply each term by $e^{- \cos (x)}$ .

$e^{- \cos (x)} \frac{d y}{d x} + e^{- \cos (x)} ((\sin (x)) y) = e^{- \cos (x)} e^{\cos (x)}$

Step 2.2

Remove parentheses.

$e^{- \cos (x)} \frac{d y}{d x} + e^{- \cos (x)} \sin (x) y = e^{- \cos (x)} e^{\cos (x)}$

Step 2.3

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

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Step 2.3.1

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

$e^{- \cos (x)} \frac{d y}{d x} + e^{- \cos (x)} \sin (x) y = e^{- \cos (x) + \cos (x)}$

Step 2.3.2

Add $- \cos (x)$ and $\cos (x)$ .

$e^{- \cos (x)} \frac{d y}{d x} + e^{- \cos (x)} \sin (x) y = e^{0}$

Step 2.4

Simplify $e^{0}$ .

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

Step 2.5

Reorder factors in $e^{- \cos (x)} \frac{d y}{d x} + e^{- \cos (x)} \sin (x) y = 1$ .

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

Step 3

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

$\frac{d}{d x} [e^{- \cos (x)} y] = 1$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{- \cos (x)} y] d x = \int d x$

Step 5

Integrate the left side.

$e^{- \cos (x)} y = \int d x$

Step 6

Apply the constant rule.

$e^{- \cos (x)} y = x + C$

Step 7

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

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Step 7.1

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

$\frac{e^{- \cos (x)} y}{e^{- \cos (x)}} = \frac{x}{e^{- \cos (x)}} + \frac{C}{e^{- \cos (x)}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $e^{- \cos (x)}$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{e^{- \cos (x)} y}{e^{- \cos (x)}} = \frac{x}{e^{- \cos (x)}} + \frac{C}{e^{- \cos (x)}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{e^{- \cos (x)}} + \frac{C}{e^{- \cos (x)}}$