Calculus Examples

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

Step 1

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

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Step 1.1

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

$\frac{d y}{d x} + y \frac{1}{x} = \cos (x)$

Step 1.2

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

$\frac{d y}{d x} + \frac{1}{x} y = \cos (x)$

Step 2

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

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Step 2.1

Set up the integration.

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

Step 2.2

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

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

Step 2.3

Remove the constant of integration.

$e^{\ln (x)}$

Step 2.4

Exponentiation and log are inverse functions.

$x$

Step 3

Multiply each term by the integrating factor $x$ .

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Step 3.1

Multiply each term by $x$ .

$x \frac{d y}{d x} + x (\frac{1}{x} y) = x \cos (x)$

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor of $x$ .

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Step 3.2.2.1

Cancel the common factor.

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

Step 3.2.2.2

Rewrite the expression.

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

Step 4

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

$\frac{d}{d x} [x y] = x \cos (x)$

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

$x y = \int x \cos (x) d x$

Step 7

Integrate the right side.

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Step 7.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \cos (x)$ .

$x y = x \sin (x) - \int \sin (x) d x$

Step 7.2

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

$x y = x \sin (x) - (- \cos (x) + C)$

Step 7.3

Rewrite $x \sin (x) - (- \cos (x) + C)$ as $x \sin (x) + \cos (x) + C$ .

$x y = x \sin (x) + \cos (x) + C$

Step 8

Divide each term in $x y = x \sin (x) + \cos (x) + C$ by $x$ and simplify.

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Step 8.1

Divide each term in $x y = x \sin (x) + \cos (x) + C$ by $x$ .

$\frac{x y}{x} = \frac{x \sin (x)}{x} + \frac{\cos (x)}{x} + \frac{C}{x}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $x$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{x y}{x} = \frac{x \sin (x)}{x} + \frac{\cos (x)}{x} + \frac{C}{x}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{x \sin (x)}{x} + \frac{\cos (x)}{x} + \frac{C}{x}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Cancel the common factor of $x$ .

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Step 8.3.1.1

Cancel the common factor.

$y = \frac{x \sin (x)}{x} + \frac{\cos (x)}{x} + \frac{C}{x}$

Step 8.3.1.2

Divide $\sin (x)$ by $1$ .

$y = \sin (x) + \frac{\cos (x)}{x} + \frac{C}{x}$