Calculus Examples

$\frac{d y}{d x} = \frac{y}{x} - \csc (\frac{y}{x})$

Step 1

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

$\frac{d y}{d x} = V - \csc (V)$

Step 2

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

$y = V x$

Step 3

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 4

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

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

Step 5

Solve the substituted differential equation.

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Step 5.1

Separate the variables.

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Step 5.1.1

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

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Step 5.1.1.1

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

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Step 5.1.1.1.1

Subtract $V$ from both sides of the equation.

$x \frac{d V}{d x} = V - \csc (V) - V$

Step 5.1.1.1.2

Combine the opposite terms in $V - \csc (V) - V$ .

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Step 5.1.1.1.2.1

Subtract $V$ from $V$ .

$x \frac{d V}{d x} = 0 - \csc (V)$

Step 5.1.1.1.2.2

Subtract $\csc (V)$ from $0$ .

$x \frac{d V}{d x} = - \csc (V)$

Step 5.1.1.2

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

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Step 5.1.1.2.1

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

$\frac{x \frac{d V}{d x}}{x} = \frac{- \csc (V)}{x}$

Step 5.1.1.2.2

Simplify the left side.

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Step 5.1.1.2.2.1

Cancel the common factor of $x$ .

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Step 5.1.1.2.2.1.1

Cancel the common factor.

$\frac{x \frac{d V}{d x}}{x} = \frac{- \csc (V)}{x}$

Step 5.1.1.2.2.1.2

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

$\frac{d V}{d x} = \frac{- \csc (V)}{x}$

Step 5.1.1.2.3

Simplify the right side.

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Step 5.1.1.2.3.1

Move the negative in front of the fraction.

$\frac{d V}{d x} = - \frac{\csc (V)}{x}$

Step 5.1.2

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

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

Step 5.1.3

Simplify.

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Step 5.1.3.1

Rewrite using the commutative property of multiplication.

$\frac{1}{\csc (V)} \frac{d V}{d x} = - \frac{1}{\csc (V)} \cdot \frac{\csc (V)}{x}$

Step 5.1.3.2

Cancel the common factor of $\csc (V)$ .

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Step 5.1.3.2.1

Move the leading negative in $- \frac{1}{\csc (V)}$ into the numerator.

$\frac{1}{\csc (V)} \frac{d V}{d x} = \frac{- 1}{\csc (V)} \cdot \frac{\csc (V)}{x}$

Step 5.1.3.2.2

Cancel the common factor.

$\frac{1}{\csc (V)} \frac{d V}{d x} = \frac{- 1}{\csc (V)} \cdot \frac{\csc (V)}{x}$

Step 5.1.3.2.3

Rewrite the expression.

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

Step 5.1.4

Rewrite the equation.

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

Step 5.2

Integrate both sides.

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Step 5.2.1

Set up an integral on each side.

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

Step 5.2.2

Integrate the left side.

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Step 5.2.2.1

Simplify.

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Step 5.2.2.1.1

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

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

Step 5.2.2.1.2

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

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

Step 5.2.2.1.3

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

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

Step 5.2.2.2

The integral of $\sin (V)$ with respect to $V$ is $- \cos (V)$ .

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

Step 5.2.3

Integrate the right side.

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Step 5.2.3.1

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

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

Step 5.2.3.2

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

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

Step 5.2.3.3

Simplify.

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

Step 5.2.4

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

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

Step 5.3

Solve for $V$ .

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Step 5.3.1

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

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Step 5.3.1.1

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

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

Step 5.3.1.2

Simplify the left side.

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Step 5.3.1.2.1

Dividing two negative values results in a positive value.

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

Step 5.3.1.2.2

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

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

Step 5.3.1.3

Simplify the right side.

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Step 5.3.1.3.1

Simplify each term.

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Step 5.3.1.3.1.1

Dividing two negative values results in a positive value.

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

Step 5.3.1.3.1.2

Divide $\ln (| x |)$ by $1$ .

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

Step 5.3.1.3.1.3

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

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

Step 5.3.1.3.1.4

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

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

Step 5.3.2

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

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

Step 5.4

Simplify the constant of integration.

$V = \arccos (\ln (| x |) + C)$

Step 6

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

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

Step 7

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

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Step 7.1

Multiply both sides by $x$ .

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

Step 7.2

Simplify.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

Cancel the common factor of $x$ .

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Step 7.2.1.1.1

Cancel the common factor.

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

Step 7.2.1.1.2

Rewrite the expression.

$y = \arccos (\ln (| x |) + C) x$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

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

$y = x \arccos (\ln (| x |) + C)$