Calculus Examples

$y' \tan (x) = 2 y - 8$

Step 1

Rewrite the differential equation.

$\frac{d y}{d x} \tan (x) = 2 y - 8$

Step 2

Separate the variables.

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Step 2.1

Divide each term in $\frac{d y}{d x} \tan (x) = 2 y - 8$ by $\tan (x)$ and simplify.

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Step 2.1.1

Divide each term in $\frac{d y}{d x} \tan (x) = 2 y - 8$ by $\tan (x)$ .

$\frac{\frac{d y}{d x} \tan (x)}{\tan (x)} = \frac{2 y}{\tan (x)} + \frac{- 8}{\tan (x)}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

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

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{\frac{d y}{d x} \tan (x)}{\tan (x)} = \frac{2 y}{\tan (x)} + \frac{- 8}{\tan (x)}$

Step 2.1.2.1.2

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

$\frac{d y}{d x} = \frac{2 y}{\tan (x)} + \frac{- 8}{\tan (x)}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Separate fractions.

$\frac{d y}{d x} = \frac{2 y}{\tan (x)} + \frac{- 8}{1} \cdot \frac{1}{\tan (x)}$

Step 2.1.3.1.2

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

$\frac{d y}{d x} = \frac{2 y}{\tan (x)} + \frac{- 8}{1} \cdot \frac{1}{\frac{\sin (x)}{\cos (x)}}$

Step 2.1.3.1.3

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

$\frac{d y}{d x} = \frac{2 y}{\tan (x)} + \frac{- 8}{1} \cdot \frac{\cos (x)}{\sin (x)}$

Step 2.1.3.1.4

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

$\frac{d y}{d x} = \frac{2 y}{\tan (x)} + \frac{- 8}{1} \cot (x)$

Step 2.1.3.1.5

Divide $- 8$ by $1$ .

$\frac{d y}{d x} = \frac{2 y}{\tan (x)} - 8 \cot (x)$

Step 2.2

Factor.

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Step 2.2.1

Factor $2$ out of $\frac{2 y}{\tan (x)} - 8 \cot (x)$ .

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Step 2.2.1.1

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

$\frac{d y}{d x} = 2 \frac{y}{\tan (x)} - 8 \cot (x)$

Step 2.2.1.2

Factor $2$ out of $- 8 \cot (x)$ .

$\frac{d y}{d x} = 2 \frac{y}{\tan (x)} + 2 (- 4 \cot (x))$

Step 2.2.1.3

Factor $2$ out of $2 \frac{y}{\tan (x)} + 2 (- 4 \cot (x))$ .

$\frac{d y}{d x} = 2 (\frac{y}{\tan (x)} - 4 \cot (x))$

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Separate fractions.

$\frac{d y}{d x} = 2 (\frac{y}{1} \cdot \frac{1}{\tan (x)} - 4 \cot (x))$

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.2.4

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

$\frac{d y}{d x} = 2 (\frac{y}{1} \cot (x) - 4 \cot (x))$

Step 2.2.2.5

Divide $y$ by $1$ .

$\frac{d y}{d x} = 2 (y \cot (x) - 4 \cot (x))$

Step 2.2.3

Factor.

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Step 2.2.3.1

Factor $\cot (x)$ out of $y \cot (x) - 4 \cot (x)$ .

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Step 2.2.3.1.1

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

$\frac{d y}{d x} = 2 (\cot (x) y - 4 \cot (x))$

Step 2.2.3.1.2

Factor $\cot (x)$ out of $- 4 \cot (x)$ .

$\frac{d y}{d x} = 2 (\cot (x) y + \cot (x) \cdot - 4)$

Step 2.2.3.1.3

Factor $\cot (x)$ out of $\cot (x) y + \cot (x) \cdot - 4$ .

$\frac{d y}{d x} = 2 (\cot (x) (y - 4))$

Step 2.2.3.2

Remove unnecessary parentheses.

$\frac{d y}{d x} = 2 \cot (x) (y - 4)$

Step 2.3

Multiply both sides by $\frac{1}{y - 4}$ .

$\frac{1}{y - 4} \frac{d y}{d x} = \frac{1}{y - 4} (2 \cot (x) (y - 4))$

Step 2.4

Simplify.

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Step 2.4.1

Rewrite using the commutative property of multiplication.

$\frac{1}{y - 4} \frac{d y}{d x} = 2 \frac{1}{y - 4} (\cot (x) (y - 4))$

Step 2.4.2

Combine $2$ and $\frac{1}{y - 4}$ .

$\frac{1}{y - 4} \frac{d y}{d x} = \frac{2}{y - 4} (\cot (x) (y - 4))$

Step 2.4.3

Cancel the common factor of $y - 4$ .

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Step 2.4.3.1

Factor $y - 4$ out of $\cot (x) (y - 4)$ .

$\frac{1}{y - 4} \frac{d y}{d x} = \frac{2}{y - 4} ((y - 4) \cot (x))$

Step 2.4.3.2

Cancel the common factor.

$\frac{1}{y - 4} \frac{d y}{d x} = \frac{2}{y - 4} ((y - 4) \cot (x))$

Step 2.4.3.3

Rewrite the expression.

$\frac{1}{y - 4} \frac{d y}{d x} = 2 \cot (x)$

Step 2.5

Rewrite the equation.

$\frac{1}{y - 4} d y = 2 \cot (x) d x$

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

$\int \frac{1}{y - 4} d y = \int 2 \cot (x) d x$

Step 3.2

Integrate the left side.

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Step 3.2.1

Let $u = y - 4$ . Then $d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.2.1.1

Let $u = y - 4$ . Find $\frac{d u}{d y}$ .

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Step 3.2.1.1.1

Differentiate $y - 4$ .

$\frac{d}{d y} [y - 4]$

Step 3.2.1.1.2

By the Sum Rule, the derivative of $y - 4$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [- 4]$ .

$\frac{d}{d y} [y] + \frac{d}{d y} [- 4]$

Step 3.2.1.1.3

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

$1 + \frac{d}{d y} [- 4]$

Step 3.2.1.1.4

Since $- 4$ is constant with respect to $y$ , the derivative of $- 4$ with respect to $y$ is $0$ .

$1 + 0$

Step 3.2.1.1.5

Add $1$ and $0$ .

$1$

Step 3.2.1.2

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

$\int \frac{1}{u} d u = \int 2 \cot (x) d x$

Step 3.2.2

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$\ln (| u |) + C_{1} = \int 2 \cot (x) d x$

Step 3.2.3

Replace all occurrences of $u$ with $y - 4$ .

$\ln (| y - 4 |) + C_{1} = \int 2 \cot (x) d x$

Step 3.3

Integrate the right side.

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Step 3.3.1

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

$\ln (| y - 4 |) + C_{1} = 2 \int \cot (x) d x$

Step 3.3.2

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

$\ln (| y - 4 |) + C_{1} = 2 (\ln (| \sin (x) |) + C_{2})$

Step 3.3.3

Simplify.

$\ln (| y - 4 |) + C_{1} = 2 \ln (| \sin (x) |) + C_{2}$

Step 3.4

Group the constant of integration on the right side as $C$ .

$\ln (| y - 4 |) = 2 \ln (| \sin (x) |) + C$

Step 4

Solve for $y$ .

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Step 4.1

Move all the terms containing a logarithm to the left side of the equation.

$\ln (| y - 4 |) - 2 \ln (| \sin (x) |) = C$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $\ln (| y - 4 |) - 2 \ln (| \sin (x) |)$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Simplify $- 2 \ln (| \sin (x) |)$ by moving $2$ inside the logarithm.

$\ln (| y - 4 |) - \ln ({| \sin (x) |}^{2}) = C$

Step 4.2.1.1.2

Remove the absolute value in ${| \sin (x) |}^{2}$ because exponentiations with even powers are always positive.

$\ln (| y - 4 |) - \ln (\sin^{2} (x)) = C$

Step 4.2.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{| y - 4 |}{\sin^{2} (x)}) = C$

Step 4.2.1.3

Multiply by $1$ .

$\ln (\frac{| y - 4 |}{\sin^{2} (x) \cdot 1}) = C$

Step 4.2.1.4

Separate fractions.

$\ln (\frac{| y - 4 |}{1} \cdot \frac{1}{\sin^{2} (x)}) = C$

Step 4.2.1.5

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

$\ln (\frac{| y - 4 |}{1} \csc^{2} (x)) = C$

Step 4.2.1.6

Divide $| y - 4 |$ by $1$ .

$\ln (| y - 4 | \csc^{2} (x)) = C$

Step 4.3

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (| y - 4 | \csc^{2} (x))} = e^{C}$

Step 4.4

Rewrite $\ln (| y - 4 | \csc^{2} (x)) = C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{C} = | y - 4 | \csc^{2} (x)$

Step 4.5

Solve for $y$ .

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Step 4.5.1

Rewrite the equation as $| y - 4 | \csc^{2} (x) = e^{C}$ .

$| y - 4 | \csc^{2} (x) = e^{C}$

Step 4.5.2

Divide each term in $| y - 4 | \csc^{2} (x) = e^{C}$ by $\csc^{2} (x)$ and simplify.

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Step 4.5.2.1

Divide each term in $| y - 4 | \csc^{2} (x) = e^{C}$ by $\csc^{2} (x)$ .

$\frac{| y - 4 | \csc^{2} (x)}{\csc^{2} (x)} = \frac{e^{C}}{\csc^{2} (x)}$

Step 4.5.2.2

Simplify the left side.

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Step 4.5.2.2.1

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

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Step 4.5.2.2.1.1

Cancel the common factor.

$\frac{| y - 4 | \csc^{2} (x)}{\csc^{2} (x)} = \frac{e^{C}}{\csc^{2} (x)}$

Step 4.5.2.2.1.2

Divide $| y - 4 |$ by $1$ .

$| y - 4 | = \frac{e^{C}}{\csc^{2} (x)}$

Step 4.5.3

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$y - 4 = \pm \frac{e^{C}}{\csc^{2} (x)}$

Step 4.5.4

Add $4$ to both sides of the equation.

$y = \pm \frac{e^{C}}{\csc^{2} (x)} + 4$

Step 5

Group the constant terms together.

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Step 5.1

Simplify the constant of integration.

$y = \pm \frac{C}{\csc^{2} (x)} + 4$

Step 5.2

Combine constants with the plus or minus.

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