Calculus Examples

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

Step 1

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

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Step 1.1

Divide each term in $\cos (x) \frac{d y}{d x} + \sin (x) y = 1$ by $\cos (x)$ .

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

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 1.3

Factor $y$ out of $\frac{\sin (x) y}{\cos (x)}$ .

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

Step 1.4

Reorder $y$ and $\frac{\sin (x)}{\cos (x)}$ .

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

Step 2

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

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Step 2.1

Set up the integration.

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

Step 2.2

Integrate $\frac{\sin (x)}{\cos (x)}$ .

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Step 2.2.1

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

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

Step 2.2.2

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

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

Step 2.3

Remove the constant of integration.

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

Step 2.4

Exponentiation and log are inverse functions.

$\sec (x)$

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

Multiply $\frac{1}{\cos (x)} \cdot \frac{\sin (x) y}{\cos (x)}$ .

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Step 3.2.5.1

Multiply $\frac{1}{\cos (x)}$ by $\frac{\sin (x) y}{\cos (x)}$ .

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

Step 3.2.5.2

Raise $\cos (x)$ to the power of $1$ .

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

Step 3.2.5.3

Raise $\cos (x)$ to the power of $1$ .

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

Step 3.2.5.4

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

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

Step 3.2.5.5

Add $1$ and $1$ .

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

Step 3.3

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

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

Step 3.4

Multiply $\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)}$ .

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Step 3.4.1

Multiply $\frac{1}{\cos (x)}$ by $\frac{1}{\cos (x)}$ .

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

Step 3.4.2

Raise $\cos (x)$ to the power of $1$ .

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

Step 3.4.3

Raise $\cos (x)$ to the power of $1$ .

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

Step 3.4.4

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

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

Step 3.4.5

Add $1$ and $1$ .

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

Step 3.5

Simplify each term.

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Step 3.5.1

Separate fractions.

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

Step 3.5.2

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

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

Step 3.5.3

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 3.5.4

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

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

Step 3.5.5

Separate fractions.

$\frac{d y}{d x} \sec (x) + \frac{y}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} = \frac{1}{\cos^{2} (x)}$

Step 3.5.6

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

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

Step 3.5.7

Separate fractions.

$\frac{d y}{d x} \sec (x) + \frac{y}{1} \cdot \frac{1}{\cos (x)} \tan (x) = \frac{1}{\cos^{2} (x)}$

Step 3.5.8

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

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

Step 3.5.9

Divide $y$ by $1$ .

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

Step 3.6

Rewrite $1$ as $1^{2}$ .

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

Step 3.7

Rewrite $\frac{1^{2}}{\cos^{2} (x)}$ as ${(\frac{1}{\cos (x)})}^{2}$ .

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

Step 3.8

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

$\frac{d y}{d x} \sec (x) + y \sec (x) \tan (x) = \sec^{2} (x)$

Step 3.9

Reorder factors in $\frac{d y}{d x} \sec (x) + y \sec (x) \tan (x) = \sec^{2} (x)$ .

$\sec (x) \frac{d y}{d x} + y \sec (x) \tan (x) = \sec^{2} (x)$

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

$\sec (x) y = \int \sec^{2} (x) d x$

Step 7

Since the derivative of $\tan (x)$ is $\sec^{2} (x)$ , the integral of $\sec^{2} (x)$ is $\tan (x)$ .

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

Step 8

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

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Step 8.1

Divide each term in $\sec (x) y = \tan (x) + C$ by $\sec (x)$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

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

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

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

Step 8.3.1.2

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

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

Step 8.3.1.3

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

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

Step 8.3.1.4

Write $\cos (x)$ as a fraction with denominator $1$ .

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

Step 8.3.1.5

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

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Step 8.3.1.5.1

Cancel the common factor.

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

Step 8.3.1.5.2

Rewrite the expression.

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

Step 8.3.1.6

Separate fractions.

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

Step 8.3.1.7

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

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

Step 8.3.1.8

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

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

Step 8.3.1.9

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

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

Step 8.3.1.10

Divide $C$ by $1$ .

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