Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

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

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Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

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

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

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

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

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Step 1.1.3.1.1.1

Factor $x$ out of $- x y$ .

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

Step 1.1.3.1.1.2

Cancel the common factors.

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Step 1.1.3.1.1.2.1

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

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

Step 1.1.3.1.1.2.2

Cancel the common factor.

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

Step 1.1.3.1.1.2.3

Rewrite the expression.

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

Step 1.1.3.1.2

Move the negative in front of the fraction.

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

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

Move $x^{- 2}$ .

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

Step 2.3.3.2.2

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

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Step 2.3.3.2.2.1

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

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

Step 2.3.3.2.2.2

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

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

Step 2.3.3.2.3

Add $- 2$ and $1$ .

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

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

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

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

Step 2.3.7

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 2.3.8

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

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

Step 3.3

To multiply absolute values, multiply the terms inside each absolute value.

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Solve for $y$ .

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Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

Divide each term in $y x = \pm e^{- \frac{1}{x} + C}$ by $x$ and simplify.

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Step 3.6.3.1

Divide each term in $y x = \pm e^{- \frac{1}{x} + C}$ by $x$ .

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

Step 3.6.3.2

Simplify the left side.

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Step 3.6.3.2.1

Cancel the common factor of $x$ .

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Step 3.6.3.2.1.1

Cancel the common factor.

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

Step 3.6.3.2.1.2

Divide $y$ by $1$ .

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

Step 3.6.3.3

Simplify the right side.

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Step 3.6.3.3.1

Simplify the numerator.

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Step 3.6.3.3.1.1

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

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

Step 3.6.3.3.1.2

Combine the numerators over the common denominator.

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