Calculus Examples

$\cos^{2} (x) \sin (x) \frac{d y}{d x} + (\cos^{3} (x)) y = 1$

Step 1

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

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Step 1.1

Reorder terms.

$\cos^{2} (x) \sin (x) \frac{d y}{d x} + y \cos^{3} (x) = 1$

Step 1.2

Factor $y$ out of $y \cos^{3} (x)$ .

$\cos^{2} (x) \sin (x) \frac{d y}{d x} + y (\cos^{3} (x)) = 1$

Step 1.3

Reorder $y$ and $\cos^{3} (x)$ .

$\cos^{2} (x) \sin (x) \frac{d y}{d x} + \cos^{3} (x) y = 1$

Step 1.4

Divide each term in $\cos^{2} (x) \sin (x) \frac{d y}{d x} + \cos^{3} (x) y = 1$ by $\cos^{2} (x) \sin (x)$ .

$\frac{\cos^{2} (x) \sin (x) \frac{d y}{d x}}{\cos^{2} (x) \sin (x)} + \frac{\cos^{3} (x) y}{\cos^{2} (x) \sin (x)} = \frac{1}{\cos^{2} (x) \sin (x)}$

Step 1.5

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

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Step 1.5.1

Cancel the common factor.

$\frac{\cos^{2} (x) \sin (x) \frac{d y}{d x}}{\cos^{2} (x) \sin (x)} + \frac{\cos^{3} (x) y}{\cos^{2} (x) \sin (x)} = \frac{1}{\cos^{2} (x) \sin (x)}$

Step 1.5.2

Rewrite the expression.

$\frac{\sin (x) \frac{d y}{d x}}{\sin (x)} + \frac{\cos^{3} (x) y}{\cos^{2} (x) \sin (x)} = \frac{1}{\cos^{2} (x) \sin (x)}$

Step 1.6

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

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Step 1.6.1

Cancel the common factor.

$\frac{\sin (x) \frac{d y}{d x}}{\sin (x)} + \frac{\cos^{3} (x) y}{\cos^{2} (x) \sin (x)} = \frac{1}{\cos^{2} (x) \sin (x)}$

Step 1.6.2

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

$\frac{d y}{d x} + \frac{\cos^{3} (x) y}{\cos^{2} (x) \sin (x)} = \frac{1}{\cos^{2} (x) \sin (x)}$

Step 1.7

Cancel the common factor of $\cos^{3} (x)$ and $\cos^{2} (x)$ .

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Step 1.7.1

Factor $\cos^{2} (x)$ out of $\cos^{3} (x) y$ .

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

Step 1.7.2

Cancel the common factors.

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Step 1.7.2.1

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

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

Step 1.7.2.2

Cancel the common factor.

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

Step 1.7.2.3

Rewrite the expression.

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

Step 1.8

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

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

Step 1.9

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

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

Step 2

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

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Step 2.1

Set up the integration.

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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 $\sin (x)$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

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

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Step 3.2.2.1

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

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

Step 3.2.2.2

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

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

Step 3.2.2.3

Cancel the common factors.

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

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.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

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

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

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

Step 8.3.1.2

Rewrite $\frac{\frac{\sin (x)}{\cos (x)}}{\sin (x)}$ as a product.

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

Step 8.3.1.3

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

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Step 8.3.1.3.1

Cancel the common factor.

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

Step 8.3.1.3.2

Rewrite the expression.

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

Step 8.3.2

Simplify each term.

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Step 8.3.2.1

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

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

Step 8.3.2.2

Separate fractions.

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

Step 8.3.2.3

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

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

Step 8.3.2.4

Divide $C$ by $1$ .

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