Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Split the fraction $\frac{e^{x} - y}{x}$ into two fractions.

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

Step 1.1.2

Subtract $\frac{- y}{x}$ from both sides of the equation.

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

Step 1.2

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

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

Step 1.3

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

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

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

Simplify.

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Step 2.2.1.1

Move the negative in front of the fraction.

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

Step 2.2.1.2

Multiply $- 1$ by $- 1$ .

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

Step 2.2.1.3

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

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

Step 2.2.2

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

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

Step 2.3

Remove the constant of integration.

$e^{\ln (x)}$

Step 2.4

Exponentiation and log are inverse functions.

$x$

Step 3

Multiply each term by the integrating factor $x$ .

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Step 3.1

Multiply each term by $x$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 3.2.3

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

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

Step 3.2.4

Cancel the common factor of $x$ .

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Step 3.2.4.1

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

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

Step 3.2.4.2

Factor $x$ out of $- x$ .

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

Step 3.2.4.3

Cancel the common factor.

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

Step 3.2.4.4

Rewrite the expression.

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

Step 3.2.5

Multiply $- 1$ by $- 1$ .

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

Step 3.2.6

Multiply $y$ by $1$ .

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

Step 3.3

Cancel the common factor of $x$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

$x y = \int e^{x} d x$

Step 7

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

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

Step 8

Divide each term in $x y = e^{x} + C$ by $x$ and simplify.

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Step 8.1

Divide each term in $x y = e^{x} + C$ by $x$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $x$ .

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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