Calculus Examples

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

Step 1

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

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Step 1.1

Set up the integration.

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

Step 1.2

Apply the constant rule.

$e^{- x + C}$

Step 1.3

Remove the constant of integration.

$e^{- x}$

Step 2

Multiply each term by the integrating factor $e^{- x}$ .

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Step 2.1

Multiply each term by $e^{- x}$ .

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

Multiply $e^{- x}$ by $e^{2 x}$ by adding the exponents.

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Step 2.3.1

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

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

Step 2.3.2

Add $- x$ and $2 x$ .

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

Step 2.4

Reorder factors in $e^{- x} \frac{d y}{d x} - e^{- x} y = e^{x}$ .

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

Step 3

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

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

Step 4

Set up an integral on each side.

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

Step 5

Integrate the left side.

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $e^{- x}$ .

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $y$ by $1$ .

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

Step 7.3

Simplify the right side.

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Step 7.3.1

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

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Step 7.3.1.1

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

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

Step 7.3.1.2

Cancel the common factors.

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Step 7.3.1.2.1

Multiply by $1$ .

$y = \frac{e^{- x} e^{2 x}}{e^{- x} \cdot 1} + \frac{C}{e^{- x}}$

Step 7.3.1.2.2

Cancel the common factor.

$y = \frac{e^{- x} e^{2 x}}{e^{- x} \cdot 1} + \frac{C}{e^{- x}}$

Step 7.3.1.2.3

Rewrite the expression.

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

Step 7.3.1.2.4

Divide $e^{2 x}$ by $1$ .

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