Calculus Examples

$\frac{d y}{d x} = x - 2 y \cot (2 x)$

Step 1

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

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Step 1.1

Add $2 y \cot (2 x)$ to both sides of the equation.

$\frac{d y}{d x} + 2 y \cot (2 x) = x$

Step 1.2

Factor $y$ out of $2 y \cot (2 x)$ .

$\frac{d y}{d x} + y (2 \cot (2 x)) = x$

Step 1.3

Reorder $y$ and $2 \cot (2 x)$ .

$\frac{d y}{d x} + 2 \cot (2 x) y = x$

Step 2

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

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Step 2.1

Set up the integration.

$e^{\int 2 \cot (2 x) d x}$

Step 2.2

Integrate $2 \cot (2 x)$ .

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Step 2.2.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$e^{2 \int \cot (2 x) d x}$

Step 2.2.2

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.2.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Step 2.2.2.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 2.2.2.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 2.2.2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 2.2.2.1.4

Multiply $2$ by $1$ .

$2$

Step 2.2.2.2

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

$e^{2 \int \cot (u) \frac{1}{2} d u}$

Step 2.2.3

Combine $\cot (u)$ and $\frac{1}{2}$ .

$e^{2 \int \frac{\cot (u)}{2} d u}$

Step 2.2.4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$e^{2 (\frac{1}{2} \int \cot (u) d u)}$

Step 2.2.5

Simplify.

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Step 2.2.5.1

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

$e^{\frac{2}{2} \int \cot (u) d u}$

Step 2.2.5.2

Cancel the common factor of $2$ .

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Step 2.2.5.2.1

Cancel the common factor.

$e^{\frac{2}{2} \int \cot (u) d u}$

Step 2.2.5.2.2

Rewrite the expression.

$e^{1 \int \cot (u) d u}$

Step 2.2.5.3

Multiply $\int \cot (u) d u$ by $1$ .

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

Step 2.2.6

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

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

Step 2.2.7

Replace all occurrences of $u$ with $2 x$ .

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

Step 2.3

Remove the constant of integration.

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

Step 2.4

Exponentiation and log are inverse functions.

$\sin (2 x)$

Step 3

Multiply each term by the integrating factor $\sin (2 x)$ .

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Step 3.1

Multiply each term by $\sin (2 x)$ .

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

Step 3.2

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

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Step 3.2.1

Move parentheses.

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

Step 3.2.2

Reorder $\sin (2 x)$ and $2 \cot (2 x)$ .

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

Step 3.2.3

Add parentheses.

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

Step 3.2.4

Rewrite $\sin (2 x) (2 \cot (2 x) y)$ in terms of sines and cosines.

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

Step 3.2.5

Cancel the common factors.

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

Step 3.3

Reorder factors in $\sin (2 x) \frac{d y}{d x} + 2 \cos (2 x) y = \sin (2 x) x$ .

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

Step 4

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

$\frac{d}{d x} [\sin (2 x) y] = x \sin (2 x)$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\sin (2 x) y] d x = \int x \sin (2 x) d x$

Step 6

Integrate the left side.

$\sin (2 x) y = \int x \sin (2 x) d x$

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 $d v = \sin (2 x)$ .

$\sin (2 x) y = x (- \frac{1}{2} \cos (2 x)) - \int - \frac{1}{2} \cos (2 x) d x$

Step 7.2

Simplify.

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Step 7.2.1

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

$\sin (2 x) y = x (- \frac{\cos (2 x)}{2}) - \int - \frac{1}{2} \cos (2 x) d x$

Step 7.2.2

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

$\sin (2 x) y = - \frac{x \cos (2 x)}{2} - \int - \frac{1}{2} \cos (2 x) d x$

Step 7.3

Since $- \frac{1}{2}$ is constant with respect to $x$ , move $- \frac{1}{2}$ out of the integral.

$\sin (2 x) y = - \frac{x \cos (2 x)}{2} - (- \frac{1}{2} \int \cos (2 x) d x)$

Step 7.4

Simplify.

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Step 7.4.1

Multiply $- 1$ by $- 1$ .

$\sin (2 x) y = - \frac{x \cos (2 x)}{2} + 1 (\frac{1}{2} \int \cos (2 x) d x)$

Step 7.4.2

Multiply $\frac{1}{2}$ by $1$ .

$\sin (2 x) y = - \frac{x \cos (2 x)}{2} + \frac{1}{2} \int \cos (2 x) d x$

Step 7.5

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.5.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Step 7.5.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 7.5.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 7.5.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 7.5.1.4

Multiply $2$ by $1$ .

$2$

Step 7.5.2

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

$\sin (2 x) y = - \frac{x \cos (2 x)}{2} + \frac{1}{2} \int \cos (u) \frac{1}{2} d u$

Step 7.6

Combine $\cos (u)$ and $\frac{1}{2}$ .

$\sin (2 x) y = - \frac{x \cos (2 x)}{2} + \frac{1}{2} \int \frac{\cos (u)}{2} d u$

Step 7.7

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\sin (2 x) y = - \frac{x \cos (2 x)}{2} + \frac{1}{2} (\frac{1}{2} \int \cos (u) d u)$

Step 7.8

Simplify.

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Step 7.8.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$\sin (2 x) y = - \frac{x \cos (2 x)}{2} + \frac{1}{2 \cdot 2} \int \cos (u) d u$

Step 7.8.2

Multiply $2$ by $2$ .

$\sin (2 x) y = - \frac{x \cos (2 x)}{2} + \frac{1}{4} \int \cos (u) d u$

Step 7.9

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

$\sin (2 x) y = - \frac{x \cos (2 x)}{2} + \frac{1}{4} (\sin (u) + C)$

Step 7.10

Simplify.

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Step 7.10.1

Rewrite $- \frac{x \cos (2 x)}{2} + \frac{1}{4} (\sin (u) + C)$ as $- \frac{1}{2} x \cos (2 x) + \frac{1}{4} \sin (u) + C$ .

$\sin (2 x) y = - \frac{1}{2} x \cos (2 x) + \frac{1}{4} \sin (u) + C$

Step 7.10.2

Simplify.

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Step 7.10.2.1

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

$\sin (2 x) y = - \frac{x}{2} \cos (2 x) + \frac{1}{4} \sin (u) + C$

Step 7.10.2.2

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

$\sin (2 x) y = - \frac{\cos (2 x) x}{2} + \frac{1}{4} \sin (u) + C$

Step 7.11

Replace all occurrences of $u$ with $2 x$ .

$\sin (2 x) y = - \frac{\cos (2 x) x}{2} + \frac{1}{4} \sin (2 x) + C$

Step 7.12

Reorder factors in $\frac{\cos (2 x) x}{2}$ .

$\sin (2 x) y = - \frac{1}{2} x \cos (2 x) + \frac{1}{4} \sin (2 x) + C$

Step 8

Solve for $y$ .

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Step 8.1

Simplify.

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Step 8.1.1

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

$\sin (2 x) y = - \frac{x}{2} \cdot \cos (2 x) + \frac{1}{4} \cdot \sin (2 x) + C$

Step 8.1.2

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

$\sin (2 x) y = - \frac{\cos (2 x) x}{2} + \frac{1}{4} \cdot \sin (2 x) + C$

Step 8.2

Divide each term in $\sin (2 x) y = - \frac{\cos (2 x) x}{2} + \frac{1}{4} \cdot \sin (2 x) + C$ by $\sin (2 x)$ and simplify.

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Step 8.2.1

Divide each term in $\sin (2 x) y = - \frac{\cos (2 x) x}{2} + \frac{1}{4} \cdot \sin (2 x) + C$ by $\sin (2 x)$ .

$\frac{\sin (2 x) y}{\sin (2 x)} = \frac{- \frac{\cos (2 x) x}{2}}{\sin (2 x)} + \frac{\frac{1}{4} \cdot \sin (2 x)}{\sin (2 x)} + \frac{C}{\sin (2 x)}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $\sin (2 x)$ .

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{\sin (2 x) y}{\sin (2 x)} = \frac{- \frac{\cos (2 x) x}{2}}{\sin (2 x)} + \frac{\frac{1}{4} \cdot \sin (2 x)}{\sin (2 x)} + \frac{C}{\sin (2 x)}$

Step 8.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{\cos (2 x) x}{2}}{\sin (2 x)} + \frac{\frac{1}{4} \cdot \sin (2 x)}{\sin (2 x)} + \frac{C}{\sin (2 x)}$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Simplify each term.

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Step 8.2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{\cos (2 x) x}{2} \cdot \frac{1}{\sin (2 x)} + \frac{\frac{1}{4} \cdot \sin (2 x)}{\sin (2 x)} + \frac{C}{\sin (2 x)}$

Step 8.2.3.1.2

Convert from $\frac{1}{\sin (2 x)}$ to $\csc (2 x)$ .

$y = - \frac{\cos (2 x) x}{2} \csc (2 x) + \frac{\frac{1}{4} \cdot \sin (2 x)}{\sin (2 x)} + \frac{C}{\sin (2 x)}$

Step 8.2.3.1.3

Combine $\csc (2 x)$ and $\frac{\cos (2 x) x}{2}$ .

$y = - \frac{\csc (2 x) (\cos (2 x) x)}{2} + \frac{\frac{1}{4} \cdot \sin (2 x)}{\sin (2 x)} + \frac{C}{\sin (2 x)}$

Step 8.2.3.1.4

Cancel the common factor of $\sin (2 x)$ .

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Step 8.2.3.1.4.1

Cancel the common factor.

$y = - \frac{\csc (2 x) \cos (2 x) x}{2} + \frac{\frac{1}{4} \cdot \sin (2 x)}{\sin (2 x)} + \frac{C}{\sin (2 x)}$

Step 8.2.3.1.4.2

Divide $\frac{1}{4}$ by $1$ .

$y = - \frac{\csc (2 x) \cos (2 x) x}{2} + \frac{1}{4} + \frac{C}{\sin (2 x)}$

Step 8.2.3.1.5

Separate fractions.

$y = - \frac{\csc (2 x) \cos (2 x) x}{2} + \frac{1}{4} + \frac{C}{1} \cdot \frac{1}{\sin (2 x)}$

Step 8.2.3.1.6

Convert from $\frac{1}{\sin (2 x)}$ to $\csc (2 x)$ .

$y = - \frac{\csc (2 x) \cos (2 x) x}{2} + \frac{1}{4} + \frac{C}{1} \csc (2 x)$

Step 8.2.3.1.7

Divide $C$ by $1$ .

$y = - \frac{\csc (2 x) \cos (2 x) x}{2} + \frac{1}{4} + C \csc (2 x)$

Step 8.2.3.2

Reorder factors in $- \frac{\csc (2 x) \cos (2 x) x}{2} + \frac{1}{4} + C \csc (2 x)$ .

$y = - \frac{x \csc (2 x) \cos (2 x)}{2} + \frac{1}{4} + C \csc (2 x)$