Calculus Examples

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

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 $y \cot (x)$ .

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

Step 1.2

Reorder $y$ and $\cot (x)$ .

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

Step 2

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

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Step 2.1

Set up the integration.

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

Step 2.2

The integral of $\cot (x)$ with respect to $x$ is $\ln (| \sin (x) |)$ .

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

Step 2.3

Remove the constant of integration.

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

Step 2.4

Exponentiation and log are inverse functions.

$\sin (x)$

Step 3

Multiply each term by the integrating factor $\sin (x)$ .

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Step 3.1

Multiply each term by $\sin (x)$ .

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

Step 3.2

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 3.2.1

Reorder $\sin (x)$ and $\cot (x)$ .

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

Step 3.2.2

Rewrite $\sin (x) (\cot (x) y)$ in terms of sines and cosines.

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

Step 3.2.3

Cancel the common factors.

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

Step 3.3

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Let $u = \sin (x)$ . Then $d u = \cos (x) d x$ , so $\frac{1}{\cos (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.1.1

Let $u = \sin (x)$ . Find $\frac{d u}{d x}$ .

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Step 7.1.1.1

Differentiate $\sin (x)$ .

$\frac{d}{d x} [\sin (x)]$

Step 7.1.1.2

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

$\cos (x)$

Step 7.1.2

Rewrite the problem using $u$ and $d u$ .

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

Step 7.2

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

$\sin (x) y = \frac{1}{2} u^{2} + C$

Step 7.3

Replace all occurrences of $u$ with $\sin (x)$ .

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $\sin (x)$ .

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

Cancel the common factor of $\sin^{2} (x)$ and $\sin (x)$ .

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Step 8.3.1.1.1

Factor $\sin (x)$ out of $\frac{1}{2} \sin^{2} (x)$ .

$y = \frac{\sin (x) (\frac{1}{2} \sin (x))}{\sin (x)} + \frac{C}{\sin (x)}$

Step 8.3.1.1.2

Cancel the common factors.

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Step 8.3.1.1.2.1

Multiply by $1$ .

$y = \frac{\sin (x) (\frac{1}{2} \sin (x))}{\sin (x) \cdot 1} + \frac{C}{\sin (x)}$

Step 8.3.1.1.2.2

Cancel the common factor.

$y = \frac{\sin (x) (\frac{1}{2} \sin (x))}{\sin (x) \cdot 1} + \frac{C}{\sin (x)}$

Step 8.3.1.1.2.3

Rewrite the expression.

$y = \frac{\frac{1}{2} \sin (x)}{1} + \frac{C}{\sin (x)}$

Step 8.3.1.1.2.4

Divide $\frac{1}{2} \sin (x)$ by $1$ .

$y = \frac{1}{2} \sin (x) + \frac{C}{\sin (x)}$

Step 8.3.1.2

Combine $\frac{1}{2}$ and $\sin (x)$ .

$y = \frac{\sin (x)}{2} + \frac{C}{\sin (x)}$

Step 8.3.1.3

Separate fractions.

$y = \frac{\sin (x)}{2} + \frac{C}{1} \cdot \frac{1}{\sin (x)}$

Step 8.3.1.4

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$y = \frac{\sin (x)}{2} + \frac{C}{1} \csc (x)$

Step 8.3.1.5

Divide $C$ by $1$ .

$y = \frac{\sin (x)}{2} + C \csc (x)$