Calculus Examples

$\frac{2 x}{y} d x - \frac{x^{2}}{y^{2}} d y = 0$

Step 1

Subtract $\frac{2 x}{y} d x$ from both sides of the equation.

$- \frac{x^{2}}{y^{2}} d y = - \frac{2 x}{y} d x$

Step 2

Multiply both sides by $\frac{y}{x^{2}}$ .

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

Step 3

Simplify.

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Step 3.1

Rewrite using the commutative property of multiplication.

$- \frac{y}{x^{2}} \cdot \frac{x^{2}}{y^{2}} d y = \frac{y}{x^{2}} (- \frac{2 x}{y}) d x$

Step 3.2

Cancel the common factor of $y$ .

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Step 3.2.1

Move the leading negative in $- \frac{y}{x^{2}}$ into the numerator.

$\frac{- y}{x^{2}} \cdot \frac{x^{2}}{y^{2}} d y = \frac{y}{x^{2}} (- \frac{2 x}{y}) d x$

Step 3.2.2

Factor $y$ out of $- y$ .

$\frac{y \cdot - 1}{x^{2}} \cdot \frac{x^{2}}{y^{2}} d y = \frac{y}{x^{2}} (- \frac{2 x}{y}) d x$

Step 3.2.3

Factor $y$ out of $y^{2}$ .

$\frac{y \cdot - 1}{x^{2}} \cdot \frac{x^{2}}{y \cdot y} d y = \frac{y}{x^{2}} (- \frac{2 x}{y}) d x$

Step 3.2.4

Cancel the common factor.

$\frac{y \cdot - 1}{x^{2}} \cdot \frac{x^{2}}{y \cdot y} d y = \frac{y}{x^{2}} (- \frac{2 x}{y}) d x$

Step 3.2.5

Rewrite the expression.

$\frac{- 1}{x^{2}} \cdot \frac{x^{2}}{y} d y = \frac{y}{x^{2}} (- \frac{2 x}{y}) d x$

Step 3.3

Cancel the common factor of $x^{2}$ .

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Step 3.3.1

Cancel the common factor.

$\frac{- 1}{x^{2}} \cdot \frac{x^{2}}{y} d y = \frac{y}{x^{2}} (- \frac{2 x}{y}) d x$

Step 3.3.2

Rewrite the expression.

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

Step 3.4

Rewrite using the commutative property of multiplication.

$- \frac{1}{y} d y = - \frac{y}{x^{2}} \cdot \frac{2 x}{y} d x$

Step 3.5

Cancel the common factor of $y$ .

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Step 3.5.1

Move the leading negative in $- \frac{y}{x^{2}}$ into the numerator.

$- \frac{1}{y} d y = \frac{- y}{x^{2}} \cdot \frac{2 x}{y} d x$

Step 3.5.2

Factor $y$ out of $- y$ .

$- \frac{1}{y} d y = \frac{y \cdot - 1}{x^{2}} \cdot \frac{2 x}{y} d x$

Step 3.5.3

Cancel the common factor.

$- \frac{1}{y} d y = \frac{y \cdot - 1}{x^{2}} \cdot \frac{2 x}{y} d x$

Step 3.5.4

Rewrite the expression.

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

Step 3.6

Cancel the common factor of $x$ .

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Step 3.6.1

Factor $x$ out of $x^{2}$ .

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

Step 3.6.2

Factor $x$ out of $2 x$ .

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

Step 3.6.3

Cancel the common factor.

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

Step 3.6.4

Rewrite the expression.

$- \frac{1}{y} d y = \frac{- 1}{x} \cdot 2 d x$

Step 3.7

Combine $\frac{- 1}{x}$ and $2$ .

$- \frac{1}{y} d y = \frac{- 1 \cdot 2}{x} d x$

Step 3.8

Multiply $- 1$ by $2$ .

$- \frac{1}{y} d y = \frac{- 2}{x} d x$

Step 3.9

Move the negative in front of the fraction.

$- \frac{1}{y} d y = - \frac{2}{x} d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int - \frac{1}{y} d y = \int - \frac{2}{x} d x$

Step 4.2

Integrate the left side.

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Step 4.2.1

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

$- \int \frac{1}{y} d y = \int - \frac{2}{x} d x$

Step 4.2.2

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

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

Step 4.2.3

Simplify.

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

Step 4.3

Integrate the right side.

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

Multiply $2$ by $- 1$ .

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

Step 4.3.4

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

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

Step 4.3.5

Simplify.

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

Step 4.4

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

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

Step 5

Solve for $y$ .

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Step 5.1

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

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

Step 5.2.1.2

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

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

Step 5.3

Subtract $\ln (x^{2})$ from both sides of the equation.

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

Step 5.4

Divide each term in $- \ln (| y |) = C - \ln (x^{2})$ by $- 1$ and simplify.

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Step 5.4.1

Divide each term in $- \ln (| y |) = C - \ln (x^{2})$ by $- 1$ .

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Dividing two negative values results in a positive value.

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

Step 5.4.2.2

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

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

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Simplify each term.

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Step 5.4.3.1.1

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

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

Step 5.4.3.1.2

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

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

Step 5.4.3.1.3

Dividing two negative values results in a positive value.

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

Step 5.4.3.1.4

Divide $\ln (x^{2})$ by $1$ .

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

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

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

Step 5.8

Rewrite $\ln (\frac{| y |}{x^{2}}) = - C$ 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 $b^{y} = x$ .

$e^{- C} = \frac{| y |}{x^{2}}$

Step 5.9

Solve for $y$ .

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Step 5.9.1

Rewrite the equation as $\frac{| y |}{x^{2}} = e^{- C}$ .

$\frac{| y |}{x^{2}} = e^{- C}$

Step 5.9.2

Multiply both sides by $x^{2}$ .

$\frac{| y |}{x^{2}} x^{2} = e^{- C} x^{2}$

Step 5.9.3

Simplify.

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Step 5.9.3.1

Simplify the left side.

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Step 5.9.3.1.1

Cancel the common factor of $x^{2}$ .

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Step 5.9.3.1.1.1

Cancel the common factor.

$\frac{| y |}{x^{2}} x^{2} = e^{- C} x^{2}$

Step 5.9.3.1.1.2

Rewrite the expression.

$| y | = e^{- C} x^{2}$

Step 5.9.3.2

Simplify the right side.

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Step 5.9.3.2.1

Reorder factors in $e^{- C} x^{2}$ .

$| y | = x^{2} e^{- C}$

Step 5.9.4

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

$y = \pm x^{2} e^{- C}$

Step 6

Group the constant terms together.

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Step 6.1

Simplify the constant of integration.

$y = \pm x^{2} C$

Step 6.2

Combine constants with the plus or minus.

$y = C x^{2}$