Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $x$ .

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Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

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

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

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Cancel the common factor of $x$ .

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Step 1.1.3.1.1

Cancel the common factor.

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

Step 1.1.3.1.2

Divide $y$ by $1$ .

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

Step 1.2

Factor.

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Step 1.2.1

Factor $y$ out of $\frac{y}{x} + y$ .

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Step 1.2.1.1

Factor $y$ out of $\frac{y}{x}$ .

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

Step 1.2.1.2

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

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

Step 1.2.1.3

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

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

Step 1.2.1.4

Factor $y$ out of $y \frac{1}{x} + y \cdot 1$ .

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

Step 1.2.2

Write $1$ as a fraction with a common denominator.

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

Step 1.2.3

Combine the numerators over the common denominator.

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

Step 1.3

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

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

Step 1.4

Cancel the common factor of $y$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

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

Step 1.5

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the fraction into multiple fractions.

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

Step 2.3.2

Split the single integral into multiple integrals.

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

Step 2.3.3

Cancel the common factor of $x$ .

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Step 2.3.3.1

Cancel the common factor.

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

Step 2.3.3.2

Rewrite the expression.

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

Step 2.3.4

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

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

Step 2.3.5

Apply the constant rule.

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

Step 2.3.6

Simplify.

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

Step 2.3.7

Reorder terms.

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

Step 2.4

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

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

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.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Solve for $y$ .

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Step 3.5.1

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

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

Step 3.5.2

Multiply both sides by $| x |$ .

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

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.

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

Step 3.5.3.1.2

Rewrite the expression.

$| y | = e^{x + C} | x |$

Step 3.5.4

Solve for $y$ .

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Step 3.5.4.1

Reorder factors in $e^{x + C} | x |$ .

$| y | = | x | e^{x + C}$

Step 3.5.4.2

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

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

Step 4

Group the constant terms together.

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Step 4.1

Rewrite $e^{x + C}$ as $e^{x} e^{C}$ .

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

Step 4.2

Reorder $e^{x}$ and $e^{C}$ .

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

Step 4.3

Combine constants with the plus or minus.

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