Calculus Examples

$y' - 2 y = 0$

Step 1

Rewrite the differential equation.

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

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 $e^{- 2 x}$ .

$e^{- 2 x} \frac{d y}{d x} + e^{- 2 x} (- 2 y) = e^{- 2 x} \cdot 0$

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Multiply $e^{- 2 x}$ by $0$ .

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

Step 3.4

Reorder factors in $e^{- 2 x} \frac{d y}{d x} - 2 e^{- 2 x} y = 0$ .

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

The integral of $0$ with respect to $x$ is $0$ .

$e^{- 2 x} y = 0 + C$

Step 7.2

Add $0$ and $C$ .

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

Step 8

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

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Step 8.1

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

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

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.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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