Calculus Examples

$\cot (x) \frac{d y}{d x} - 2 y = 5$

Step 1

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

Tap for more steps...

Step 1.1

Divide each term in $\cot (x) \frac{d y}{d x} - 2 y = 5$ by $\cot (x)$ .

$\frac{\cot (x) \frac{d y}{d x}}{\cot (x)} + \frac{- 2 y}{\cot (x)} = \frac{5}{\cot (x)}$

Step 1.2

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

Tap for more steps...

Step 1.2.1

Cancel the common factor.

$\frac{\cot (x) \frac{d y}{d x}}{\cot (x)} + \frac{- 2 y}{\cot (x)} = \frac{5}{\cot (x)}$

Step 1.2.2

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

$\frac{d y}{d x} + \frac{- 2 y}{\cot (x)} = \frac{5}{\cot (x)}$

Step 1.3

Factor $y$ out of $\frac{- 2 y}{\cot (x)}$ .

$\frac{d y}{d x} + y \frac{- 2}{\cot (x)} = \frac{5}{\cot (x)}$

Step 1.4

Reorder $y$ and $\frac{- 2}{\cot (x)}$ .

$\frac{d y}{d x} + \frac{- 2}{\cot (x)} y = \frac{5}{\cot (x)}$

Step 2

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

Tap for more steps...

Step 2.1

Set up the integration.

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

Step 2.2

Integrate $\frac{- 2}{\cot (x)}$ .

Tap for more steps...

Step 2.2.1

Move the negative in front of the fraction.

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Simplify.

Tap for more steps...

Step 2.2.4.1

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

$e^{- (2 \int \frac{1}{\frac{\cos (x)}{\sin (x)}} d x)}$

Step 2.2.4.2

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

$e^{- (2 \int \frac{\sin (x)}{\cos (x)} d x)}$

Step 2.2.4.3

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

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

Step 2.2.4.4

Multiply $2$ by $- 1$ .

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

Step 2.2.5

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

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

Step 2.2.6

Simplify.

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

Step 2.3

Remove the constant of integration.

$e^{- 2 \ln (\sec (x))}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (\sec^{- 2} (x))}$

Step 2.5

Exponentiation and log are inverse functions.

$\sec^{- 2} (x)$

Step 2.6

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

$\frac{1}{\sec^{2} (x)}$

Step 2.7

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

$\frac{1^{2}}{\sec^{2} (x)}$

Step 2.8

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

${(\frac{1}{\sec (x)})}^{2}$

Step 2.9

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

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

Step 2.10

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

${(1 \cos (x))}^{2}$

Step 2.11

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

$\cos^{2} (x)$

Step 3

Multiply each term by the integrating factor $\cos^{2} (x)$ .

Tap for more steps...

Step 3.1

Multiply each term by $\cos^{2} (x)$ .

$\cos^{2} (x) \frac{d y}{d x} + \cos^{2} (x) (\frac{- 2}{\cot (x)} y) = \cos^{2} (x) \frac{5}{\cot (x)}$

Step 3.2

Simplify each term.

Tap for more steps...

Step 3.2.1

Separate fractions.

$\cos^{2} (x) \frac{d y}{d x} + \cos^{2} (x) (\frac{- 2}{1} \cdot \frac{1}{\cot (x)} y) = \cos^{2} (x) \frac{5}{\cot (x)}$

Step 3.2.2

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

$\cos^{2} (x) \frac{d y}{d x} + \cos^{2} (x) (\frac{- 2}{1} \cdot \frac{1}{\frac{\cos (x)}{\sin (x)}} y) = \cos^{2} (x) \frac{5}{\cot (x)}$

Step 3.2.3

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

$\cos^{2} (x) \frac{d y}{d x} + \cos^{2} (x) (\frac{- 2}{1} (1 \frac{\sin (x)}{\cos (x)}) y) = \cos^{2} (x) \frac{5}{\cot (x)}$

Step 3.2.4

Write $1$ as a fraction with denominator $1$ .

$\cos^{2} (x) \frac{d y}{d x} + \cos^{2} (x) (\frac{- 2}{1} (\frac{1}{1} \cdot \frac{\sin (x)}{\cos (x)}) y) = \cos^{2} (x) \frac{5}{\cot (x)}$

Step 3.2.5

Simplify.

Tap for more steps...

Step 3.2.5.1

Rewrite the expression.

$\cos^{2} (x) \frac{d y}{d x} + \cos^{2} (x) (\frac{- 2}{1} (1 \frac{\sin (x)}{\cos (x)}) y) = \cos^{2} (x) \frac{5}{\cot (x)}$

Step 3.2.5.2

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

$\cos^{2} (x) \frac{d y}{d x} + \cos^{2} (x) (\frac{- 2}{1} \cdot \frac{\sin (x)}{\cos (x)} y) = \cos^{2} (x) \frac{5}{\cot (x)}$

Step 3.2.6

Rewrite using the commutative property of multiplication.

$\cos^{2} (x) \frac{d y}{d x} + \frac{- 2}{1} \cos^{2} (x) \frac{\sin (x)}{\cos (x)} y = \cos^{2} (x) \frac{5}{\cot (x)}$

Step 3.2.7

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

Tap for more steps...

Step 3.2.7.1

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

$\cos^{2} (x) \frac{d y}{d x} + \cos (x) (\frac{- 2}{1} \cos (x)) \frac{\sin (x)}{\cos (x)} y = \cos^{2} (x) \frac{5}{\cot (x)}$

Step 3.2.7.2

Cancel the common factor.

$\cos^{2} (x) \frac{d y}{d x} + \cos (x) (\frac{- 2}{1} \cos (x)) \frac{\sin (x)}{\cos (x)} y = \cos^{2} (x) \frac{5}{\cot (x)}$

Step 3.2.7.3

Rewrite the expression.

$\cos^{2} (x) \frac{d y}{d x} + \frac{- 2}{1} \cos (x) \sin (x) y = \cos^{2} (x) \frac{5}{\cot (x)}$

Step 3.2.8

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

$\cos^{2} (x) \frac{d y}{d x} + \frac{- 2 \cos (x)}{1} \sin (x) y = \cos^{2} (x) \frac{5}{\cot (x)}$

Step 3.2.9

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

$\cos^{2} (x) \frac{d y}{d x} + \frac{- 2 \cos (x) \sin (x)}{1} y = \cos^{2} (x) \frac{5}{\cot (x)}$

Step 3.2.10

Divide $- 2 \cos (x) \sin (x)$ by $1$ .

$\cos^{2} (x) \frac{d y}{d x} - 2 \cos (x) \sin (x) y = \cos^{2} (x) \frac{5}{\cot (x)}$

Step 3.3

Separate fractions.

$\cos^{2} (x) \frac{d y}{d x} - 2 \cos (x) \sin (x) y = \cos^{2} (x) (\frac{5}{1} \cdot \frac{1}{\cot (x)})$

Step 3.4

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

$\cos^{2} (x) \frac{d y}{d x} - 2 \cos (x) \sin (x) y = \cos^{2} (x) (\frac{5}{1} \cdot \frac{1}{\frac{\cos (x)}{\sin (x)}})$

Step 3.5

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

$\cos^{2} (x) \frac{d y}{d x} - 2 \cos (x) \sin (x) y = \cos^{2} (x) (\frac{5}{1} (1 \frac{\sin (x)}{\cos (x)}))$

Step 3.6

Write $1$ as a fraction with denominator $1$ .

$\cos^{2} (x) \frac{d y}{d x} - 2 \cos (x) \sin (x) y = \cos^{2} (x) (\frac{5}{1} (\frac{1}{1} \cdot \frac{\sin (x)}{\cos (x)}))$

Step 3.7

Simplify.

Tap for more steps...

Step 3.7.1

Rewrite the expression.

$\cos^{2} (x) \frac{d y}{d x} - 2 \cos (x) \sin (x) y = \cos^{2} (x) (\frac{5}{1} (1 \frac{\sin (x)}{\cos (x)}))$

Step 3.7.2

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

$\cos^{2} (x) \frac{d y}{d x} - 2 \cos (x) \sin (x) y = \cos^{2} (x) (\frac{5}{1} \cdot \frac{\sin (x)}{\cos (x)})$

Step 3.8

Rewrite using the commutative property of multiplication.

$\cos^{2} (x) \frac{d y}{d x} - 2 \cos (x) \sin (x) y = \frac{5}{1} \cos^{2} (x) \frac{\sin (x)}{\cos (x)}$

Step 3.9

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

Tap for more steps...

Step 3.9.1

Factor $\cos (x)$ out of $\frac{5}{1} \cos^{2} (x)$ .

$\cos^{2} (x) \frac{d y}{d x} - 2 \cos (x) \sin (x) y = \cos (x) (\frac{5}{1} \cos (x)) \frac{\sin (x)}{\cos (x)}$

Step 3.9.2

Cancel the common factor.

$\cos^{2} (x) \frac{d y}{d x} - 2 \cos (x) \sin (x) y = \cos (x) (\frac{5}{1} \cos (x)) \frac{\sin (x)}{\cos (x)}$

Step 3.9.3

Rewrite the expression.

$\cos^{2} (x) \frac{d y}{d x} - 2 \cos (x) \sin (x) y = \frac{5}{1} \cos (x) \sin (x)$

Step 3.10

Combine $\frac{5}{1}$ and $\cos (x)$ .

$\cos^{2} (x) \frac{d y}{d x} - 2 \cos (x) \sin (x) y = \frac{5 \cos (x)}{1} \sin (x)$

Step 3.11

Combine $\frac{5 \cos (x)}{1}$ and $\sin (x)$ .

$\cos^{2} (x) \frac{d y}{d x} - 2 \cos (x) \sin (x) y = \frac{5 \cos (x) \sin (x)}{1}$

Step 3.12

Divide $5 \cos (x) \sin (x)$ by $1$ .

$\cos^{2} (x) \frac{d y}{d x} - 2 \cos (x) \sin (x) y = 5 \cos (x) \sin (x)$

Step 3.13

Reorder factors in $\cos^{2} (x) \frac{d y}{d x} - 2 \cos (x) \sin (x) y = 5 \cos (x) \sin (x)$ .

$\cos^{2} (x) \frac{d y}{d x} - 2 y \cos (x) \sin (x) = 5 \cos (x) \sin (x)$

Step 4

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

$\frac{d}{d x} [\cos^{2} (x) y] = 5 \cos (x) \sin (x)$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\cos^{2} (x) y] d x = \int 5 \cos (x) \sin (x) d x$

Step 6

Integrate the left side.

$\cos^{2} (x) y = \int 5 \cos (x) \sin (x) d x$

Step 7

Integrate the right side.

Tap for more steps...

Step 7.1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$\cos^{2} (x) y = 5 \int \cos (x) \sin (x) d x$

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 $d$ $u$ .

Tap for more steps...

Step 7.2.1

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

Tap for more steps...

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 $d u$ .

$\cos^{2} (x) y = 5 \int - u d u$

Step 7.3

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

$\cos^{2} (x) y = 5 (- \int u d u)$

Step 7.4

Multiply $- 1$ by $5$ .

$\cos^{2} (x) y = - 5 \int u d u$

Step 7.5

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

$\cos^{2} (x) y = - 5 (\frac{1}{2} u^{2} + C)$

Step 7.6

Simplify.

Tap for more steps...

Step 7.6.1

Rewrite $- 5 (\frac{1}{2} u^{2} + C)$ as $- 5 (\frac{1}{2}) u^{2} + C$ .

$\cos^{2} (x) y = - 5 (\frac{1}{2}) u^{2} + C$

Step 7.6.2

Simplify.

Tap for more steps...

Step 7.6.2.1

Combine $- 5$ and $\frac{1}{2}$ .

$\cos^{2} (x) y = \frac{- 5}{2} u^{2} + C$

Step 7.6.2.2

Move the negative in front of the fraction.

$\cos^{2} (x) y = - \frac{5}{2} u^{2} + C$

Step 7.7

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

$\cos^{2} (x) y = - \frac{5}{2} \cos^{2} (x) + C$

Step 8

Divide each term in $\cos^{2} (x) y = - \frac{5}{2} \cos^{2} (x) + C$ by $\cos^{2} (x)$ and simplify.

Tap for more steps...

Step 8.1

Divide each term in $\cos^{2} (x) y = - \frac{5}{2} \cos^{2} (x) + C$ by $\cos^{2} (x)$ .

$\frac{\cos^{2} (x) y}{\cos^{2} (x)} = \frac{- \frac{5}{2} \cos^{2} (x)}{\cos^{2} (x)} + \frac{C}{\cos^{2} (x)}$

Step 8.2

Simplify the left side.

Tap for more steps...

Step 8.2.1

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

Tap for more steps...

Step 8.2.1.1

Cancel the common factor.

$\frac{\cos^{2} (x) y}{\cos^{2} (x)} = \frac{- \frac{5}{2} \cos^{2} (x)}{\cos^{2} (x)} + \frac{C}{\cos^{2} (x)}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{5}{2} \cos^{2} (x)}{\cos^{2} (x)} + \frac{C}{\cos^{2} (x)}$

Step 8.3

Simplify the right side.

Tap for more steps...

Step 8.3.1

Simplify each term.

Tap for more steps...

Step 8.3.1.1

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

Tap for more steps...

Step 8.3.1.1.1

Cancel the common factor.

$y = \frac{- \frac{5}{2} \cos^{2} (x)}{\cos^{2} (x)} + \frac{C}{\cos^{2} (x)}$

Step 8.3.1.1.2

Divide $- \frac{5}{2}$ by $1$ .

$y = - \frac{5}{2} + \frac{C}{\cos^{2} (x)}$

Step 8.3.1.2

Multiply by $1$ .

$y = - \frac{5}{2} + \frac{C}{\cos^{2} (x) \cdot 1}$

Step 8.3.1.3

Separate fractions.

$y = - \frac{5}{2} + \frac{C}{1} \cdot \frac{1}{\cos^{2} (x)}$

Step 8.3.1.4

Convert from $\frac{1}{\cos^{2} (x)}$ to $\sec^{2} (x)$ .

$y = - \frac{5}{2} + \frac{C}{1} \sec^{2} (x)$

Step 8.3.1.5

Divide $C$ by $1$ .

$y = - \frac{5}{2} + C \sec^{2} (x)$