Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $x$ .

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Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

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

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

Step 1.3

Simplify the right side.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Cancel the common factor of $x$ .

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Step 1.3.1.1.1

Cancel the common factor.

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

Step 1.3.1.1.2

Divide $\sin (\frac{y}{x}) \tan (\frac{y}{x})$ by $1$ .

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

Step 1.3.1.2

Rewrite $\tan (\frac{y}{x})$ in terms of sines and cosines.

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

Step 1.3.1.3

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

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Step 1.3.1.3.1

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

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

Step 1.3.1.3.2

Raise $\sin (\frac{y}{x})$ to the power of $1$ .

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

Step 1.3.1.3.3

Raise $\sin (\frac{y}{x})$ to the power of $1$ .

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

Step 1.3.1.3.4

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

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

Step 1.3.1.3.5

Add $1$ and $1$ .

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

Step 1.3.2

Simplify each term.

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Step 1.3.2.1

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

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

Step 1.3.2.2

Separate fractions.

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

Step 1.3.2.3

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

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

Step 1.3.2.4

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

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

Step 2

Let $V = \frac{y}{x}$ . Substitute $V$ for $\frac{y}{x}$ .

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

Step 3

Solve $V = \frac{y}{x}$ for $y$ .

$y = V x$

Step 4

Use the product rule to find the derivative of $y = V x$ with respect to $x$ .

$\frac{d y}{d x} = x \frac{d V}{d x} + V$

Step 5

Substitute $x \frac{d V}{d x} + V$ for $\frac{d y}{d x}$ .

$x \frac{d V}{d x} + V = V + \sin (V) \tan (V)$

Step 6

Solve the substituted differential equation.

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Step 6.1

Separate the variables.

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Step 6.1.1

Solve for $\frac{d V}{d x}$ .

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Step 6.1.1.1

Simplify the right side.

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Step 6.1.1.1.1

Simplify $V + \sin (V) \tan (V)$ .

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Step 6.1.1.1.1.1

Simplify each term.

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Step 6.1.1.1.1.1.1

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

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

Step 6.1.1.1.1.1.2

Multiply $\sin (V) \frac{\sin (V)}{\cos (V)}$ .

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Step 6.1.1.1.1.1.2.1

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

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

Step 6.1.1.1.1.1.2.2

Raise $\sin (V)$ to the power of $1$ .

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

Step 6.1.1.1.1.1.2.3

Raise $\sin (V)$ to the power of $1$ .

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

Step 6.1.1.1.1.1.2.4

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

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

Step 6.1.1.1.1.1.2.5

Add $1$ and $1$ .

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

Step 6.1.1.1.1.2

Simplify each term.

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Step 6.1.1.1.1.2.1

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

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

Step 6.1.1.1.1.2.2

Separate fractions.

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

Step 6.1.1.1.1.2.3

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

$x \frac{d V}{d x} + V = V + \frac{\sin (V)}{1} \tan (V)$

Step 6.1.1.1.1.2.4

Divide $\sin (V)$ by $1$ .

$x \frac{d V}{d x} + V = V + \sin (V) \tan (V)$

Step 6.1.1.2

Move all terms not containing $\frac{d V}{d x}$ to the right side of the equation.

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Step 6.1.1.2.1

Subtract $V$ from both sides of the equation.

$x \frac{d V}{d x} = V + \sin (V) \tan (V) - V$

Step 6.1.1.2.2

Combine the opposite terms in $V + \sin (V) \tan (V) - V$ .

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Step 6.1.1.2.2.1

Subtract $V$ from $V$ .

$x \frac{d V}{d x} = \sin (V) \tan (V) + 0$

Step 6.1.1.2.2.2

Add $\sin (V) \tan (V)$ and $0$ .

$x \frac{d V}{d x} = \sin (V) \tan (V)$

Step 6.1.1.2.3

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

$x \frac{d V}{d x} = \sin (V) \frac{\sin (V)}{\cos (V)}$

Step 6.1.1.2.4

Multiply $\sin (V) \frac{\sin (V)}{\cos (V)}$ .

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Step 6.1.1.2.4.1

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

$x \frac{d V}{d x} = \frac{\sin (V) \sin (V)}{\cos (V)}$

Step 6.1.1.2.4.2

Raise $\sin (V)$ to the power of $1$ .

$x \frac{d V}{d x} = \frac{\sin^{1} (V) \sin (V)}{\cos (V)}$

Step 6.1.1.2.4.3

Raise $\sin (V)$ to the power of $1$ .

$x \frac{d V}{d x} = \frac{\sin^{1} (V) \sin^{1} (V)}{\cos (V)}$

Step 6.1.1.2.4.4

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

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

Step 6.1.1.2.4.5

Add $1$ and $1$ .

$x \frac{d V}{d x} = \frac{\sin^{2} (V)}{\cos (V)}$

Step 6.1.1.2.5

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

$x \frac{d V}{d x} = \frac{\sin (V) \sin (V)}{\cos (V)}$

Step 6.1.1.2.6

Separate fractions.

$x \frac{d V}{d x} = \frac{\sin (V)}{1} \cdot \frac{\sin (V)}{\cos (V)}$

Step 6.1.1.2.7

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

$x \frac{d V}{d x} = \frac{\sin (V)}{1} \tan (V)$

Step 6.1.1.2.8

Divide $\sin (V)$ by $1$ .

$x \frac{d V}{d x} = \sin (V) \tan (V)$

Step 6.1.1.3

Divide each term in $x \frac{d V}{d x} = \sin (V) \tan (V)$ by $x$ and simplify.

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Step 6.1.1.3.1

Divide each term in $x \frac{d V}{d x} = \sin (V) \tan (V)$ by $x$ .

$\frac{x \frac{d V}{d x}}{x} = \frac{\sin (V) \tan (V)}{x}$

Step 6.1.1.3.2

Simplify the left side.

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Step 6.1.1.3.2.1

Cancel the common factor of $x$ .

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Step 6.1.1.3.2.1.1

Cancel the common factor.

$\frac{x \frac{d V}{d x}}{x} = \frac{\sin (V) \tan (V)}{x}$

Step 6.1.1.3.2.1.2

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

$\frac{d V}{d x} = \frac{\sin (V) \tan (V)}{x}$

Step 6.1.2

Multiply both sides by $\frac{1}{\sin (V) \tan (V)}$ .

$\frac{1}{\sin (V) \tan (V)} \frac{d V}{d x} = \frac{1}{\sin (V) \tan (V)} \cdot \frac{\sin (V) \tan (V)}{x}$

Step 6.1.3

Cancel the common factor of $\sin (V) \tan (V)$ .

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Step 6.1.3.1

Cancel the common factor.

$\frac{1}{\sin (V) \tan (V)} \frac{d V}{d x} = \frac{1}{\sin (V) \tan (V)} \cdot \frac{\sin (V) \tan (V)}{x}$

Step 6.1.3.2

Rewrite the expression.

$\frac{1}{\sin (V) \tan (V)} \frac{d V}{d x} = \frac{1}{x}$

Step 6.1.4

Rewrite the equation.

$\frac{1}{\sin (V) \tan (V)} d V = \frac{1}{x} d x$

Step 6.2

Integrate both sides.

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Step 6.2.1

Set up an integral on each side.

$\int \frac{1}{\sin (V) \tan (V)} d V = \int \frac{1}{x} d x$

Step 6.2.2

Integrate the left side.

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Step 6.2.2.1

Simplify.

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Step 6.2.2.1.1

Separate fractions.

$\int \frac{1}{\tan (V)} \cdot \frac{1}{\sin (V)} d V = \int \frac{1}{x} d x$

Step 6.2.2.1.2

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

$\int \frac{1}{\tan (V)} \csc (V) d V = \int \frac{1}{x} d x$

Step 6.2.2.1.3

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

$\int \frac{1}{\frac{\sin (V)}{\cos (V)}} \csc (V) d V = \int \frac{1}{x} d x$

Step 6.2.2.1.4

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

$\int \frac{\cos (V)}{\sin (V)} \csc (V) d V = \int \frac{1}{x} d x$

Step 6.2.2.1.5

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

$\int \cot (V) \csc (V) d V = \int \frac{1}{x} d x$

Step 6.2.2.2

Since the derivative of $- \csc (V)$ is $\cot (V) \csc (V)$ , the integral of $\cot (V) \csc (V)$ is $- \csc (V)$ .

$- \csc (V) + C_{1} = \int \frac{1}{x} d x$

Step 6.2.3

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

$- \csc (V) + C_{1} = \ln (| x |) + C_{2}$

Step 6.2.4

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

$- \csc (V) = \ln (| x |) + C$

Step 6.3

Solve for $V$ .

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Step 6.3.1

Divide each term in $- \csc (V) = \ln (| x |) + C$ by $- 1$ and simplify.

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Step 6.3.1.1

Divide each term in $- \csc (V) = \ln (| x |) + C$ by $- 1$ .

$\frac{- \csc (V)}{- 1} = \frac{\ln (| x |)}{- 1} + \frac{C}{- 1}$

Step 6.3.1.2

Simplify the left side.

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Step 6.3.1.2.1

Dividing two negative values results in a positive value.

$\frac{\csc (V)}{1} = \frac{\ln (| x |)}{- 1} + \frac{C}{- 1}$

Step 6.3.1.2.2

Divide $\csc (V)$ by $1$ .

$\csc (V) = \frac{\ln (| x |)}{- 1} + \frac{C}{- 1}$

Step 6.3.1.3

Simplify the right side.

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Step 6.3.1.3.1

Simplify each term.

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Step 6.3.1.3.1.1

Move the negative one from the denominator of $\frac{\ln (| x |)}{- 1}$ .

$\csc (V) = - 1 \cdot \ln (| x |) + \frac{C}{- 1}$

Step 6.3.1.3.1.2

Rewrite $- 1 \cdot \ln (| x |)$ as $- \ln (| x |)$ .

$\csc (V) = - \ln (| x |) + \frac{C}{- 1}$

Step 6.3.1.3.1.3

Move the negative one from the denominator of $\frac{C}{- 1}$ .

$\csc (V) = - \ln (| x |) - 1 \cdot C$

Step 6.3.1.3.1.4

Rewrite $- 1 \cdot C$ as $- C$ .

$\csc (V) = - \ln (| x |) - C$

Step 6.3.2

Take the inverse cosecant of both sides of the equation to extract $V$ from inside the cosecant.

$V = arccsc (- \ln (| x |) - C)$

Step 6.4

Simplify the constant of integration.

$V = arccsc (- \ln (| x |) + C)$

Step 7

Substitute $\frac{y}{x}$ for $V$ .

$\frac{y}{x} = arccsc (- \ln (| x |) + C)$

Step 8

Solve $\frac{y}{x} = arccsc (- \ln (| x |) + C)$ for $y$ .

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Step 8.1

Multiply both sides by $x$ .

$\frac{y}{x} x = arccsc (- \ln (| x |) + C) x$

Step 8.2

Simplify.

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Step 8.2.1

Simplify the left side.

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Step 8.2.1.1

Cancel the common factor of $x$ .

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Step 8.2.1.1.1

Cancel the common factor.

$\frac{y}{x} x = arccsc (- \ln (| x |) + C) x$

Step 8.2.1.1.2

Rewrite the expression.

$y = arccsc (- \ln (| x |) + C) x$

Step 8.2.2

Simplify the right side.

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Step 8.2.2.1

Reorder factors in $arccsc (- \ln (| x |) + C) x$ .

$y = x arccsc (- \ln (| x |) + C)$