Calculus Examples

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

Step 1

Subtract $y$ from both sides of the equation.

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

Step 2

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

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Step 2.1

Set up the integration.

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

Step 2.2

Apply the constant rule.

$e^{- x + C}$

Step 2.3

Remove the constant of integration.

$e^{- x}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

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

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Step 3.4.1

Move $e^{x}$ .

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

Step 3.4.2

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

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

Step 3.4.3

Subtract $x$ from $x$ .

$e^{- x} \frac{d y}{d x} - e^{- x} y = 2 e^{0}$

Step 3.5

Simplify $2 e^{0}$ .

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

Step 3.6

Reorder factors in $e^{- x} \frac{d y}{d x} - e^{- x} y = 2$ .

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Apply the constant rule.

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

Step 8

Divide each term in $e^{- x} y = 2 x + C$ by $e^{- x}$ and simplify.

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Step 8.1

Divide each term in $e^{- x} y = 2 x + C$ by $e^{- x}$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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