Calculus Examples

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

Step 1

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

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

Step 2

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

$x = V y$

Step 3

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

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

Step 4

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

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

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 $\frac{d V}{d y}$ .

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Step 5.1.1.1

Move all terms not containing $\frac{d V}{d y}$ 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.

$y \frac{d V}{d y} = - V - V$

Step 5.1.1.1.2

Subtract $V$ from $- V$ .

$y \frac{d V}{d y} = - 2 V$

Step 5.1.1.2

Divide each term in $y \frac{d V}{d y} = - 2 V$ by $y$ and simplify.

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Step 5.1.1.2.1

Divide each term in $y \frac{d V}{d y} = - 2 V$ by $y$ .

$\frac{y \frac{d V}{d y}}{y} = \frac{- 2 V}{y}$

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 $y$ .

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Step 5.1.1.2.2.1.1

Cancel the common factor.

$\frac{y \frac{d V}{d y}}{y} = \frac{- 2 V}{y}$

Step 5.1.1.2.2.1.2

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

$\frac{d V}{d y} = \frac{- 2 V}{y}$

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.

$\frac{d V}{d y} = - \frac{2 V}{y}$

Step 5.1.2

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

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

Step 5.1.3

Simplify.

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Step 5.1.3.1

Rewrite using the commutative property of multiplication.

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

Step 5.1.3.2

Cancel the common factor of $V$ .

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Step 5.1.3.2.1

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

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

Step 5.1.3.2.2

Factor $V$ out of $2 V$ .

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

Step 5.1.3.2.3

Cancel the common factor.

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

Step 5.1.3.2.4

Rewrite the expression.

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

Step 5.1.4

Rewrite the equation.

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

Step 5.2

Integrate both sides.

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Step 5.2.1

Set up an integral on each side.

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

Step 5.2.2

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

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

Step 5.2.3

Integrate the right side.

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Step 5.2.3.1

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

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

Step 5.2.3.2

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

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

Step 5.2.3.3

Multiply $2$ by $- 1$ .

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

Step 5.2.3.4

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

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

Step 5.2.3.5

Simplify.

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

Step 5.2.4

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

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

Step 5.3

Solve for $V$ .

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Step 5.3.1

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

$\ln (| V |) + 2 \ln (| y |) = C$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Simplify $\ln (| V |) + 2 \ln (| y |)$ .

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Step 5.3.2.1.1

Simplify each term.

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Step 5.3.2.1.1.1

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

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

Step 5.3.2.1.1.2

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

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

Step 5.3.2.1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 5.3.2.1.3

Reorder factors in $\ln (| V | y^{2})$ .

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

Step 5.3.3

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

$e^{\ln (y^{2} | V |)} = e^{C}$

Step 5.3.4

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

$e^{C} = y^{2} | V |$

Step 5.3.5

Solve for $V$ .

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Step 5.3.5.1

Rewrite the equation as $y^{2} | V | = e^{C}$ .

$y^{2} | V | = e^{C}$

Step 5.3.5.2

Divide each term in $y^{2} | V | = e^{C}$ by $y^{2}$ and simplify.

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Step 5.3.5.2.1

Divide each term in $y^{2} | V | = e^{C}$ by $y^{2}$ .

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

Step 5.3.5.2.2

Simplify the left side.

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Step 5.3.5.2.2.1

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

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Step 5.3.5.2.2.1.1

Cancel the common factor.

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

Step 5.3.5.2.2.1.2

Divide $| V |$ by $1$ .

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

Step 5.3.5.3

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

$V = \pm \frac{e^{C}}{y^{2}}$

Step 5.4

Group the constant terms together.

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Step 5.4.1

Simplify the constant of integration.

$V = \pm \frac{C}{y^{2}}$

Step 5.4.2

Combine constants with the plus or minus.

$V = C \frac{1}{y^{2}}$

Step 6

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

$\frac{x}{y} = C \frac{1}{y^{2}}$

Step 7

Solve $\frac{x}{y} = C \frac{1}{y^{2}}$ for $x$ .

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Step 7.1

Multiply both sides by $y$ .

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

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 $y$ .

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Step 7.2.1.1.1

Cancel the common factor.

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

Step 7.2.1.1.2

Rewrite the expression.

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

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $C (\frac{1}{y^{2}}) y$ .

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Step 7.2.2.1.1

Combine $C$ and $\frac{1}{y^{2}}$ .

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

Step 7.2.2.1.2

Cancel the common factor of $y$ .

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Step 7.2.2.1.2.1

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

$x = \frac{C}{y \cdot y} y$

Step 7.2.2.1.2.2

Cancel the common factor.

$x = \frac{C}{y \cdot y} y$

Step 7.2.2.1.2.3

Rewrite the expression.

$x = \frac{C}{y}$