Calculus Examples

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

Step 1

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

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Step 1.1

Subtract $y \tan (x)$ from both sides of the equation.

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

Step 1.2

Factor $y$ out of $- y \tan (x)$ .

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

Step 1.3

Reorder $y$ and $- 1 \tan (x)$ .

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

Step 2

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

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Step 2.1

Set up the integration.

$e^{\int - 1 \tan (x) d x}$

Step 2.2

Integrate $- 1 \tan (x)$ .

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Step 2.2.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$e^{- \int \tan (x) d x}$

Step 2.2.2

The integral of $\tan (x)$ with respect to $x$ is $\ln (| \sec (x) |)$ .

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

Step 2.2.3

Simplify.

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

Step 2.3

Remove the constant of integration.

$e^{- \ln (\sec (x))}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (\sec^{- 1} (x))}$

Step 2.5

Exponentiation and log are inverse functions.

$\sec^{- 1} (x)$

Step 2.6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{\sec (x)}$

Step 2.7

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{1}{\frac{1}{\cos (x)}}$

Step 2.8

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

$1 \cos (x)$

Step 2.9

Multiply $\cos (x)$ by $1$ .

$\cos (x)$

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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Step 3.2.1.1

Move parentheses.

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

Step 3.2.1.2

Reorder $\cos (x)$ and $- 1 \tan (x)$ .

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

Step 3.2.1.3

Add parentheses.

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

Step 3.2.1.4

Rewrite $\cos (x) (- 1 \tan (x) y)$ in terms of sines and cosines.

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

Step 3.2.1.5

Cancel the common factors.

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

Step 3.2.2

Rewrite $- 1 \sin (x)$ as $- \sin (x)$ .

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

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

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

Step 7.2

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

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Step 7.2.1

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

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Step 7.2.1.1

Differentiate $\cos (x)$ .

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

Step 7.2.1.2

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

$- \sin (x)$

Step 7.2.2

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

$\cos (x) y = - 2 \int - u d u$

Step 7.3

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$\cos (x) y = - 2 (- \int u d u)$

Step 7.4

Multiply $- 1$ by $- 2$ .

$\cos (x) y = 2 \int u d u$

Step 7.5

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

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

Step 7.6

Simplify.

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Step 7.6.1

Rewrite $2 (\frac{1}{2} u^{2} + C)$ as $2 (\frac{1}{2}) u^{2} + C$ .

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

Step 7.6.2

Simplify.

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Step 7.6.2.1

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

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

Step 7.6.2.2

Cancel the common factor of $2$ .

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Step 7.6.2.2.1

Cancel the common factor.

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

Step 7.6.2.2.2

Rewrite the expression.

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

Step 7.6.2.3

Multiply $u^{2}$ by $1$ .

$\cos (x) y = u^{2} + C$

Step 7.7

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

$\cos (x) y = \cos^{2} (x) + C$

Step 8

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

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Step 8.1

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

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

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

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 $\cos^{2} (x)$ and $\cos (x)$ .

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Step 8.3.1.1.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

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

Step 8.3.1.1.2

Cancel the common factors.

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Step 8.3.1.1.2.1

Multiply by $1$ .

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

Step 8.3.1.1.2.2

Cancel the common factor.

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

Step 8.3.1.1.2.3

Rewrite the expression.

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

Step 8.3.1.1.2.4

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

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

Step 8.3.1.2

Separate fractions.

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

Step 8.3.1.3

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 8.3.1.4

Divide $C$ by $1$ .

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