Calculus Examples

$m x d y = n y d x$

Step 1

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

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

Step 2

Simplify.

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Step 2.1

Cancel the common factor of $x$ .

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Step 2.1.1

Factor $x$ out of $x y$ .

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

Step 2.1.2

Factor $x$ out of $m x$ .

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

Step 2.1.3

Cancel the common factor.

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

Step 2.1.4

Rewrite the expression.

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

Step 2.2

Combine $\frac{1}{y}$ and $m$ .

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

Step 2.3

Cancel the common factor of $y$ .

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Step 2.3.1

Factor $y$ out of $x y$ .

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

Step 2.3.2

Factor $y$ out of $n y$ .

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

Step 2.3.3

Cancel the common factor.

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

Step 2.3.4

Rewrite the expression.

$\frac{m}{y} d y = \frac{1}{x} n d x$

Step 2.4

Combine $\frac{1}{x}$ and $n$ .

$\frac{m}{y} d y = \frac{n}{x} d x$

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

$\int \frac{m}{y} d y = \int \frac{n}{x} d x$

Step 3.2

Integrate the left side.

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Step 3.2.1

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

$m \int \frac{1}{y} d y = \int \frac{n}{x} d x$

Step 3.2.2

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

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

Step 3.2.3

Simplify.

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

Step 3.3

Integrate the right side.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Simplify.

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

Step 3.4

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

$m \ln (| y |) = n \ln (| x |) + C$

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.

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

Step 4.2

Add $n \ln (| x |)$ to both sides of the equation.

$m \ln (| y |) = C + n \ln (| x |)$

Step 4.3

Divide each term in $m \ln (| y |) = C + n \ln (| x |)$ by $\ln (| y |)$ and simplify.

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Step 4.3.1

Divide each term in $m \ln (| y |) = C + n \ln (| x |)$ by $\ln (| y |)$ .

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

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $\ln (| y |)$ .

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Step 4.3.2.1.1

Cancel the common factor.

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

Step 4.3.2.1.2

Divide $m$ by $1$ .

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

Step 4.4

Rewrite the equation as $\frac{C}{\ln (| y |)} + \frac{n \ln (| x |)}{\ln (| y |)} = m$ .

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

Step 4.5

Multiply each term in $\frac{C}{\ln (| y |)} + \frac{n \ln (| x |)}{\ln (| y |)} = m$ by $\ln (| y |)$ to eliminate the fractions.

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Step 4.5.1

Multiply each term in $\frac{C}{\ln (| y |)} + \frac{n \ln (| x |)}{\ln (| y |)} = m$ by $\ln (| y |)$ .

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

Step 4.5.2

Simplify the left side.

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Step 4.5.2.1

Simplify each term.

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Step 4.5.2.1.1

Combine $\frac{C}{\ln (| y |)}$ and $\ln (| y |)$ .

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

Step 4.5.2.1.2

Combine $\frac{n \ln (| x |)}{\ln (| y |)}$ and $\ln (| y |)$ .

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

Step 4.6

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

$\frac{C \ln (| y |)}{\ln (| y |)} + \frac{n \ln (| x |) \ln (| y |)}{\ln (| y |)} - m \ln (| y |) = 0$

Step 4.7

Simplify each term.

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Step 4.7.1

Cancel the common factor of $\ln (| y |)$ .

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Step 4.7.1.1

Cancel the common factor.

$\frac{C \ln (| y |)}{\ln (| y |)} + \frac{n \ln (| x |) \ln (| y |)}{\ln (| y |)} - m \ln (| y |) = 0$

Step 4.7.1.2

Divide $C$ by $1$ .

$C + \frac{n \ln (| x |) \ln (| y |)}{\ln (| y |)} - m \ln (| y |) = 0$

Step 4.7.2

Cancel the common factor of $\ln (| y |)$ .

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Step 4.7.2.1

Cancel the common factor.

$C + \frac{n \ln (| x |) \ln (| y |)}{\ln (| y |)} - m \ln (| y |) = 0$

Step 4.7.2.2

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

$C + n \ln (| x |) - m \ln (| y |) = 0$

Step 4.8

Move all terms not containing $\ln (| y |)$ to the right side of the equation.

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Step 4.8.1

Subtract $C$ from both sides of the equation.

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

Step 4.8.2

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

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

Step 4.9

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

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Step 4.9.1

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

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

Step 4.9.2

Simplify the left side.

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Step 4.9.2.1

Dividing two negative values results in a positive value.

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

Step 4.9.2.2

Cancel the common factor of $m$ .

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Step 4.9.2.2.1

Cancel the common factor.

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

Step 4.9.2.2.2

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

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

Step 4.9.3

Simplify the right side.

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Step 4.9.3.1

Simplify each term.

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Step 4.9.3.1.1

Dividing two negative values results in a positive value.

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

Step 4.9.3.1.2

Dividing two negative values results in a positive value.

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

Step 4.10

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

$e^{\ln (| y |)} = e^{\frac{C}{m} + \frac{n \ln (| x |)}{m}}$

Step 4.11

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

$e^{\frac{C}{m} + \frac{n \ln (| x |)}{m}} = | y |$

Step 4.12

Solve for $y$ .

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Step 4.12.1

Rewrite the equation as $| y | = e^{\frac{C}{m} + \frac{n \ln (| x |)}{m}}$ .

$| y | = e^{\frac{C}{m} + \frac{n \ln (| x |)}{m}}$

Step 4.12.2

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

$y = \pm e^{\frac{C}{m} + \frac{n \ln (| x |)}{m}}$