Calculus Examples

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

Step 1

Rewrite the differential equation as $\frac{d y}{d x} + M (x) y = Q (x)$ .

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Step 1.1

Subtract $y$ from both sides of the equation.

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

Step 1.2

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

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

Step 1.3

Cancel the common factor of $x$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

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

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

Step 1.4

Cancel the common factor of $x$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Divide $- 1$ by $1$ .

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

Step 1.5

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

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

Step 1.6

Reorder $y$ and $\frac{- 1}{x}$ .

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{- 1}{x}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{- 1}{x} d x}$

Step 2.2

Integrate $\frac{- 1}{x}$ .

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Step 2.2.1

Split the fraction into multiple fractions.

$e^{\int - \frac{1}{x} d x}$

Step 2.2.2

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

$e^{- \int \frac{1}{x} d x}$

Step 2.2.3

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

$e^{- (\ln (| x |) + C)}$

Step 2.2.4

Simplify.

$e^{- \ln (| x |) + C}$

Step 2.3

Remove the constant of integration.

$e^{- \ln (x)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (x^{- 1})}$

Step 2.5

Exponentiation and log are inverse functions.

$x^{- 1}$

Step 2.6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x}$

Step 3

Multiply each term by the integrating factor $\frac{1}{x}$ .

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Step 3.1

Multiply each term by $\frac{1}{x}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $\frac{1}{x}$ and $\frac{d y}{d x}$ .

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 3.2.3

Rewrite using the commutative property of multiplication.

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

Step 3.2.4

Combine $\frac{1}{x}$ and $y$ .

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

Step 3.2.5

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

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Step 3.2.5.1

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

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

Step 3.2.5.2

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

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

Step 3.2.5.3

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

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

Step 3.2.5.4

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

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

Step 3.2.5.5

Add $1$ and $1$ .

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

Step 3.3

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

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

Step 3.4

Move the negative in front of the fraction.

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

Step 4

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [\frac{1}{x} y] = - \frac{1}{x}$

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Simplify.

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

Step 8

Solve for $y$ .

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Step 8.1

Combine $\frac{1}{x}$ and $y$ .

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

Step 8.2

Multiply both sides by $x$ .

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

Step 8.3

Simplify.

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Step 8.3.1

Simplify the left side.

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Step 8.3.1.1

Cancel the common factor of $x$ .

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Step 8.3.1.1.1

Cancel the common factor.

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

Step 8.3.1.1.2

Rewrite the expression.

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

Step 8.3.2

Simplify the right side.

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Step 8.3.2.1

Simplify $(- \ln (| x |) + C) x$ .

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Step 8.3.2.1.1

Apply the distributive property.

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

Step 8.3.2.1.2

Simplify the expression.

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Step 8.3.2.1.2.1

Reorder factors in $- \ln (| x |) x + C x$ .

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

Step 8.3.2.1.2.2

Reorder $- x \ln (| x |)$ and $C x$ .

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