Calculus Examples

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

Step 1

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

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Step 1.1

Divide each term in $e^{x} \frac{d y}{d x} + 2 e^{x} y = 1$ by $e^{x}$ .

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

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

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

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

Step 1.3

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

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Divide $2 y$ by $1$ .

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 2$ .

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Step 2.1

Set up the integration.

$e^{\int 2 d x}$

Step 2.2

Apply the constant rule.

$e^{2 x + C}$

Step 2.3

Remove the constant of integration.

$e^{2 x}$

Step 3

Multiply each term by the integrating factor $e^{2 x}$ .

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Step 3.1

Multiply each term by $e^{2 x}$ .

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Combine $e^{2 x}$ and $\frac{1}{e^{x}}$ .

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

Step 3.4

Cancel the common factor of $e^{2 x}$ and $e^{x}$ .

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Step 3.4.1

Factor $e^{x}$ out of $e^{2 x}$ .

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

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Multiply by $1$ .

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

Step 3.4.2.2

Cancel the common factor.

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

Step 3.4.2.3

Rewrite the expression.

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

Step 3.4.2.4

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

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

Step 3.5

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $e^{2 x}$ .

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Cancel the common factor of $e^{x}$ and $e^{2 x}$ .

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Step 8.3.1.1

Factor $e^{2 x}$ out of $e^{x}$ .

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

Step 8.3.1.2

Cancel the common factors.

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Step 8.3.1.2.1

Multiply by $1$ .

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

Step 8.3.1.2.2

Cancel the common factor.

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

Step 8.3.1.2.3

Rewrite the expression.

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

Step 8.3.1.2.4

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

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