Calculus Examples

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

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = - 4$ .

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Step 1.1

Set up the integration.

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

Step 1.2

Apply the constant rule.

$e^{- 4 x + C}$

Step 1.3

Remove the constant of integration.

$e^{- 4 x}$

Step 2

Multiply each term by the integrating factor $e^{- 4 x}$ .

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Step 2.1

Multiply each term by $e^{- 4 x}$ .

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

Multiply $e^{- 4 x}$ by $e^{x}$ 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.

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

Step 2.3.2

Add $- 4 x$ and $x$ .

$e^{- 4 x} \frac{d y}{d x} - 4 e^{- 4 x} y = e^{- 3 x}$

Step 2.4

Reorder factors in $e^{- 4 x} \frac{d y}{d x} - 4 e^{- 4 x} y = e^{- 3 x}$ .

$e^{- 4 x} \frac{d y}{d x} - 4 y e^{- 4 x} = e^{- 3 x}$

Step 3

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

$\frac{d}{d x} [e^{- 4 x} y] = e^{- 3 x}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{- 4 x} y] d x = \int e^{- 3 x} d x$

Step 5

Integrate the left side.

$e^{- 4 x} y = \int e^{- 3 x} d x$

Step 6

Integrate the right side.

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Step 6.1

Let $u = - 3 x$ . Then $d u = - 3 d x$ , so $- \frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1.1

Let $u = - 3 x$ . Find $\frac{d u}{d x}$ .

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Step 6.1.1.1

Differentiate $- 3 x$ .

$\frac{d}{d x} [- 3 x]$

Step 6.1.1.2

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$- 3 \frac{d}{d x} [x]$

Step 6.1.1.3

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

$- 3 \cdot 1$

Step 6.1.1.4

Multiply $- 3$ by $1$ .

$- 3$

Step 6.1.2

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

$e^{- 4 x} y = \int e^{u} \frac{1}{- 3} d u$

Step 6.2

Simplify.

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Step 6.2.1

Move the negative in front of the fraction.

$e^{- 4 x} y = \int e^{u} (- \frac{1}{3}) d u$

Step 6.2.2

Combine $e^{u}$ and $\frac{1}{3}$ .

$e^{- 4 x} y = \int - \frac{e^{u}}{3} d u$

Step 6.3

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

$e^{- 4 x} y = - \int \frac{e^{u}}{3} d u$

Step 6.4

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$e^{- 4 x} y = - (\frac{1}{3} \int e^{u} d u)$

Step 6.5

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

$e^{- 4 x} y = - \frac{1}{3} (e^{u} + C)$

Step 6.6

Simplify.

$e^{- 4 x} y = - \frac{1}{3} e^{u} + C$

Step 6.7

Replace all occurrences of $u$ with $- 3 x$ .

$e^{- 4 x} y = - \frac{1}{3} e^{- 3 x} + C$

Step 7

Divide each term in $e^{- 4 x} y = - \frac{1}{3} e^{- 3 x} + C$ by $e^{- 4 x}$ and simplify.

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Step 7.1

Divide each term in $e^{- 4 x} y = - \frac{1}{3} e^{- 3 x} + C$ by $e^{- 4 x}$ .

$\frac{e^{- 4 x} y}{e^{- 4 x}} = \frac{- \frac{1}{3} e^{- 3 x}}{e^{- 4 x}} + \frac{C}{e^{- 4 x}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $e^{- 4 x}$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{e^{- 4 x} y}{e^{- 4 x}} = \frac{- \frac{1}{3} e^{- 3 x}}{e^{- 4 x}} + \frac{C}{e^{- 4 x}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{1}{3} e^{- 3 x}}{e^{- 4 x}} + \frac{C}{e^{- 4 x}}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Cancel the common factor of $e^{- 3 x}$ and $e^{- 4 x}$ .

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Step 7.3.1.1.1

Factor $e^{- 4 x}$ out of $- \frac{1}{3} e^{- 3 x}$ .

$y = \frac{e^{- 4 x} (- \frac{1}{3} e^{x})}{e^{- 4 x}} + \frac{C}{e^{- 4 x}}$

Step 7.3.1.1.2

Cancel the common factors.

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Step 7.3.1.1.2.1

Multiply by $1$ .

$y = \frac{e^{- 4 x} (- \frac{1}{3} e^{x})}{e^{- 4 x} \cdot 1} + \frac{C}{e^{- 4 x}}$

Step 7.3.1.1.2.2

Cancel the common factor.

$y = \frac{e^{- 4 x} (- \frac{1}{3} e^{x})}{e^{- 4 x} \cdot 1} + \frac{C}{e^{- 4 x}}$

Step 7.3.1.1.2.3

Rewrite the expression.

$y = \frac{- \frac{1}{3} e^{x}}{1} + \frac{C}{e^{- 4 x}}$

Step 7.3.1.1.2.4

Divide $- \frac{1}{3} e^{x}$ by $1$ .

$y = - \frac{1}{3} e^{x} + \frac{C}{e^{- 4 x}}$

Step 7.3.1.2

Combine $e^{x}$ and $\frac{1}{3}$ .

$y = - \frac{e^{x}}{3} + \frac{C}{e^{- 4 x}}$