Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

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

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

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

Step 2.2.2

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

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

Step 2.2.3

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 $\frac{1}{x}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Rewrite using the commutative property of multiplication.

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

Step 3.2.3

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

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

Step 3.2.4

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

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Step 3.2.4.1

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

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

Step 3.2.4.2

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

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

Step 3.2.4.3

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

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

Step 3.2.4.4

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

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

Step 3.2.4.5

Add $1$ and $1$ .

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

Cancel the common factor of $x$ .

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Step 3.4.1

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

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

Step 3.4.2

Factor $x$ out of $x e^{- x}$ .

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

Step 3.4.3

Cancel the common factor.

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

Step 3.4.4

Rewrite the expression.

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

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.

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

Step 7.2

Let $u = - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.2.1

Let $u = - x$ . Find $\frac{d u}{d x}$ .

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Step 7.2.1.1

Differentiate $- x$ .

$\frac{d}{d x} [- x]$

Step 7.2.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$- \frac{d}{d x} [x]$

Step 7.2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 1 \cdot 1$

Step 7.2.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 7.2.2

Rewrite the problem using $u$ and $d u$ .

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

Step 7.3

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

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

Step 7.4

Simplify.

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Step 7.4.1

Multiply $- 1$ by $- 1$ .

$\frac{1}{x} y = 1 \int e^{u} d u$

Step 7.4.2

Multiply $\int e^{u} d u$ by $1$ .

$\frac{1}{x} y = \int e^{u} d u$

Step 7.5

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

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

Step 7.6

Replace all occurrences of $u$ with $- x$ .

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

Step 8

Solve for $y$ .

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Step 8.1

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

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

Step 8.2

Multiply both sides by $x$ .

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

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.

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

Step 8.3.1.1.2

Rewrite the expression.

$y = (e^{- x} + C) x$

Step 8.3.2

Simplify the right side.

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Step 8.3.2.1

Simplify $(e^{- x} + C) x$ .

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Step 8.3.2.1.1

Apply the distributive property.

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

Step 8.3.2.1.2

Simplify the expression.

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Step 8.3.2.1.2.1

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

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

Step 8.3.2.1.2.2

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

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