Calculus Examples

$y d x - x d y = 0$

Step 1

Substitute $y d$ for $yd$ .

$y d x - x d y = 0$

Step 2

Subtract $y d x$ from both sides of the equation.

$- x d y = - y d x$

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Rewrite using the commutative property of multiplication.

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

Step 4.2

Cancel the common factor of $x$ .

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Step 4.2.1

Move the leading negative in $- \frac{1}{x y}$ into the numerator.

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

Step 4.2.2

Factor $x$ out of $x y$ .

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

Step 4.2.3

Cancel the common factor.

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

Step 4.2.4

Rewrite the expression.

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

Step 4.3

Move the negative in front of the fraction.

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

Step 4.4

Rewrite using the commutative property of multiplication.

$- \frac{1}{y} d y = - \frac{1}{x y} y d x$

Step 4.5

Cancel the common factor of $y$ .

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Step 4.5.1

Move the leading negative in $- \frac{1}{x y}$ into the numerator.

$- \frac{1}{y} d y = \frac{- 1}{x y} y d x$

Step 4.5.2

Factor $y$ out of $x y$ .

$- \frac{1}{y} d y = \frac{- 1}{y x} y d x$

Step 4.5.3

Cancel the common factor.

$- \frac{1}{y} d y = \frac{- 1}{y x} y d x$

Step 4.5.4

Rewrite the expression.

$- \frac{1}{y} d y = \frac{- 1}{x} d x$

Step 4.6

Move the negative in front of the fraction.

$- \frac{1}{y} d y = - \frac{1}{x} d x$

Step 5

Integrate both sides.

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Step 5.1

Set up an integral on each side.

$\int - \frac{1}{y} d y = \int - \frac{1}{x} d x$

Step 5.2

Integrate the left side.

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Step 5.2.1

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

$- \int \frac{1}{y} d y = \int - \frac{1}{x} d x$

Step 5.2.2

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

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

Step 5.2.3

Simplify.

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

Step 5.3

Integrate the right side.

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Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.3.3

Simplify.

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

Step 5.4

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

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

Step 6

Solve for $y$ .

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Step 6.1

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

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

Step 6.2

Subtract $\ln (| x |)$ from both sides of the equation.

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

Step 6.3

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

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Step 6.3.1

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

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

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Dividing two negative values results in a positive value.

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

Step 6.3.2.2

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

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

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Simplify each term.

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Step 6.3.3.1.1

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

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

Step 6.3.3.1.2

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

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

Step 6.3.3.1.3

Dividing two negative values results in a positive value.

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

Step 6.3.3.1.4

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

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

Step 6.4

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

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

Step 6.5

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

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

Step 6.6

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

$e^{\ln (\frac{| y |}{| x |})} = e^{- C}$

Step 6.7

Rewrite $\ln (\frac{| y |}{| x |}) = - C$ 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 $b^{y} = x$ .

$e^{- C} = \frac{| y |}{| x |}$

Step 6.8

Solve for $y$ .

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Step 6.8.1

Rewrite the equation as $\frac{| y |}{| x |} = e^{- C}$ .

$\frac{| y |}{| x |} = e^{- C}$

Step 6.8.2

Multiply both sides by $| x |$ .

$\frac{| y |}{| x |} | x | = e^{- C} | x |$

Step 6.8.3

Simplify the left side.

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Step 6.8.3.1

Cancel the common factor of $| x |$ .

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Step 6.8.3.1.1

Cancel the common factor.

$\frac{| y |}{| x |} | x | = e^{- C} | x |$

Step 6.8.3.1.2

Rewrite the expression.

$| y | = e^{- C} | x |$

Step 6.8.4

Solve for $y$ .

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Step 6.8.4.1

Reorder factors in $e^{- C} | x |$ .

$| y | = | x | e^{- C}$

Step 6.8.4.2

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

$y = \pm | x | e^{- C}$

Step 7

Group the constant terms together.

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Step 7.1

Simplify the constant of integration.

$y = \pm | x | C$

Step 7.2

Combine constants with the plus or minus.

$y = C | x |$