Calculus Examples

$\frac{d y}{d x} - y = 4$

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 $e^{- x}$ .

$e^{- x} \frac{d y}{d x} + e^{- x} (- y) = e^{- x} \cdot 4$

Step 2.2

Rewrite using the commutative property of multiplication.

$e^{- x} \frac{d y}{d x} - e^{- x} y = e^{- x} \cdot 4$

Step 2.3

Move $4$ to the left of $e^{- x}$ .

$e^{- x} \frac{d y}{d x} - e^{- x} y = 4 \cdot e^{- x}$

Step 2.4

Reorder factors in $e^{- x} \frac{d y}{d x} - e^{- x} y = 4 e^{- x}$ .

$e^{- x} \frac{d y}{d x} - y e^{- x} = 4 e^{- x}$

Step 3

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

$\frac{d}{d x} [e^{- x} y] = 4 e^{- x}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{- x} y] d x = \int 4 e^{- x} d x$

Step 5

Integrate the left side.

$e^{- x} y = \int 4 e^{- x} d x$

Step 6

Integrate the right side.

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Step 6.1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$e^{- x} y = 4 \int e^{- x} d x$

Step 6.2

Let $u = - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.2.1

Let $u = - x$ . Find $\frac{d u}{d x}$ .

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Step 6.2.1.1

Differentiate $- x$ .

$\frac{d}{d x} [- x]$

Step 6.2.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$- \frac{d}{d x} [x]$

Step 6.2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 1 \cdot 1$

Step 6.2.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 6.2.2

Rewrite the problem using $u$ and $d u$ .

$e^{- x} y = 4 \int - e^{u} d u$

Step 6.3

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$e^{- x} y = 4 (- \int e^{u} d u)$

Step 6.4

Multiply $- 1$ by $4$ .

$e^{- x} y = - 4 \int e^{u} d u$

Step 6.5

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

$e^{- x} y = - 4 (e^{u} + C)$

Step 6.6

Simplify.

$e^{- x} y = - 4 e^{u} + C$

Step 6.7

Replace all occurrences of $u$ with $- x$ .

$e^{- x} y = - 4 e^{- x} + C$

Step 7

Divide each term in $e^{- x} y = - 4 e^{- x} + C$ by $e^{- x}$ and simplify.

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Step 7.1

Divide each term in $e^{- x} y = - 4 e^{- x} + C$ by $e^{- x}$ .

$\frac{e^{- x} y}{e^{- x}} = \frac{- 4 e^{- x}}{e^{- x}} + \frac{C}{e^{- x}}$

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.

$\frac{e^{- x} y}{e^{- x}} = \frac{- 4 e^{- x}}{e^{- x}} + \frac{C}{e^{- x}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 4 e^{- x}}{e^{- x}} + \frac{C}{e^{- x}}$

Step 7.3

Simplify the right side.

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Step 7.3.1

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

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Step 7.3.1.1

Cancel the common factor.

$y = \frac{- 4 e^{- x}}{e^{- x}} + \frac{C}{e^{- x}}$

Step 7.3.1.2

Divide $- 4$ by $1$ .

$y = - 4 + \frac{C}{e^{- x}}$