Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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Step 1.1.1

Add $y$ to both sides of the equation.

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

Step 1.1.2

Divide each term in $x \frac{d y}{d x} = \frac{y}{x} + y$ by $x$ and simplify.

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Step 1.1.2.1

Divide each term in $x \frac{d y}{d x} = \frac{y}{x} + y$ by $x$ .

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

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $x$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

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

Step 1.1.2.2.1.2

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

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

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{d y}{d x} = \frac{y}{x} \cdot \frac{1}{x} + \frac{y}{x}$

Step 1.1.2.3.1.2

Multiply $\frac{y}{x} \cdot \frac{1}{x}$ .

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Step 1.1.2.3.1.2.1

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

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

Step 1.1.2.3.1.2.2

Raise $x$ to the power of $1$ .

$\frac{d y}{d x} = \frac{y}{x^{1} x} + \frac{y}{x}$

Step 1.1.2.3.1.2.3

Raise $x$ to the power of $1$ .

$\frac{d y}{d x} = \frac{y}{x^{1} x^{1}} + \frac{y}{x}$

Step 1.1.2.3.1.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d y}{d x} = \frac{y}{x^{1 + 1}} + \frac{y}{x}$

Step 1.1.2.3.1.2.5

Add $1$ and $1$ .

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

Step 1.2

Factor.

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Step 1.2.1

To write $\frac{y}{x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 1.2.2

Write each expression with a common denominator of $x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.2.1

Multiply $\frac{y}{x}$ by $\frac{x}{x}$ .

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

Step 1.2.2.2

Raise $x$ to the power of $1$ .

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

Step 1.2.2.3

Raise $x$ to the power of $1$ .

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

Step 1.2.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.2.2.5

Add $1$ and $1$ .

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

Step 1.2.3

Combine the numerators over the common denominator.

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

Step 1.2.4

Factor $y$ out of $y + y x$ .

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Step 1.2.4.1

Raise $y$ to the power of $1$ .

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

Step 1.2.4.2

Factor $y$ out of $y^{1}$ .

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

Step 1.2.4.3

Factor $y$ out of $y x$ .

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

Step 1.2.4.4

Factor $y$ out of $y \cdot 1 + y (x)$ .

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

Step 1.2.4.5

Multiply $y$ by $1 + x$ .

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

Step 1.3

Regroup factors.

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

Step 1.4

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

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

Step 1.5

Cancel the common factor of $y$ .

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Step 1.5.1

Cancel the common factor.

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

Step 1.5.2

Rewrite the expression.

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

Step 1.6

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Apply basic rules of exponents.

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Step 2.3.1.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2.3.1.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 2.3.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.3.1.2.2

Multiply $2$ by $- 1$ .

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

Step 2.3.2

Multiply $(1 + x) x^{- 2}$ .

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

Step 2.3.3

Simplify.

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Step 2.3.3.1

Multiply $x^{- 2}$ by $1$ .

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

Step 2.3.3.2

Multiply $x$ by $x^{- 2}$ by adding the exponents.

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Step 2.3.3.2.1

Multiply $x$ by $x^{- 2}$ .

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Step 2.3.3.2.1.1

Raise $x$ to the power of $1$ .

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

Step 2.3.3.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.3.3.2.2

Subtract $2$ from $1$ .

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

Step 2.3.4

Split the single integral into multiple integrals.

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

Step 2.3.5

By the Power Rule, the integral of $x^{- 2}$ with respect to $x$ is $- x^{- 1}$ .

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

Step 2.3.6

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

$\ln (| y |) + C_{1} = - x^{- 1} + C_{2} + \ln (| x |) + C_{3}$

Step 2.3.7

Simplify.

$\ln (| y |) + C_{1} = - \frac{1}{x} + \ln (| x |) + C_{4}$

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

$e^{- \frac{1}{x} + C} = \frac{| y |}{| x |}$

Step 3.5

Solve for $y$ .

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Step 3.5.1

Rewrite the equation as $\frac{| y |}{| x |} = e^{- \frac{1}{x} + C}$ .

$\frac{| y |}{| x |} = e^{- \frac{1}{x} + C}$

Step 3.5.2

Multiply both sides by $| x |$ .

$\frac{| y |}{| x |} | x | = e^{- \frac{1}{x} + C} | x |$

Step 3.5.3

Simplify the left side.

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Step 3.5.3.1

Cancel the common factor of $| x |$ .

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Step 3.5.3.1.1

Cancel the common factor.

$\frac{| y |}{| x |} | x | = e^{- \frac{1}{x} + C} | x |$

Step 3.5.3.1.2

Rewrite the expression.

$| y | = e^{- \frac{1}{x} + C} | x |$

Step 3.5.4

Solve for $y$ .

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Step 3.5.4.1

Reorder factors in $e^{- \frac{1}{x} + C} | x |$ .

$| y | = | x | e^{- \frac{1}{x} + C}$

Step 3.5.4.2

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

$y = \pm | x | e^{- \frac{1}{x} + C}$

Step 4

Group the constant terms together.

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Step 4.1

Rewrite $e^{- \frac{1}{x} + C}$ as $e^{- \frac{1}{x}} e^{C}$ .

$y = \pm | x | (e^{- \frac{1}{x}} e^{C})$

Step 4.2

Reorder $e^{- \frac{1}{x}}$ and $e^{C}$ .

$y = \pm | x | (e^{C} e^{- \frac{1}{x}})$

Step 4.3

Combine constants with the plus or minus.

$y = C | x | e^{- \frac{1}{x}}$