Calculus Examples

$\frac{d y}{d x} + y = e^{7 x}$

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.

$e^{\int d x}$

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 $e^{x}$ .

$e^{x} \frac{d y}{d x} + e^{x} y = e^{x} e^{7 x}$

Step 2.2

Multiply $e^{x}$ by $e^{7 x}$ by adding the exponents.

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Step 2.2.1

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

$e^{x} \frac{d y}{d x} + e^{x} y = e^{x + 7 x}$

Step 2.2.2

Add $x$ and $7 x$ .

$e^{x} \frac{d y}{d x} + e^{x} y = e^{8 x}$

Step 2.3

Reorder factors in $e^{x} \frac{d y}{d x} + e^{x} y = e^{8 x}$ .

$e^{x} \frac{d y}{d x} + y e^{x} = e^{8 x}$

Step 3

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

$\frac{d}{d x} [e^{x} y] = e^{8 x}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{x} y] d x = \int e^{8 x} d x$

Step 5

Integrate the left side.

$e^{x} y = \int e^{8 x} d x$

Step 6

Integrate the right side.

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Step 6.1

Let $u = 8 x$ . Then $d u = 8 d x$ , so $\frac{1}{8} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1.1

Let $u = 8 x$ . Find $\frac{d u}{d x}$ .

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Step 6.1.1.1

Differentiate $8 x$ .

$\frac{d}{d x} [8 x]$

Step 6.1.1.2

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$8 \frac{d}{d x} [x]$

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 $n = 1$ .

$8 \cdot 1$

Step 6.1.1.4

Multiply $8$ by $1$ .

$8$

Step 6.1.2

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

$e^{x} y = \int e^{u} \frac{1}{8} d u$

Step 6.2

Combine $e^{u}$ and $\frac{1}{8}$ .

$e^{x} y = \int \frac{e^{u}}{8} d u$

Step 6.3

Since $\frac{1}{8}$ is constant with respect to $u$ , move $\frac{1}{8}$ out of the integral.

$e^{x} y = \frac{1}{8} \int e^{u} d u$

Step 6.4

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

$e^{x} y = \frac{1}{8} (e^{u} + C)$

Step 6.5

Simplify.

$e^{x} y = \frac{1}{8} e^{u} + C$

Step 6.6

Replace all occurrences of $u$ with $8 x$ .

$e^{x} y = \frac{1}{8} e^{8 x} + C$

Step 7

Divide each term in $e^{x} y = \frac{1}{8} e^{8 x} + C$ by $e^{x}$ and simplify.

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Step 7.1

Divide each term in $e^{x} y = \frac{1}{8} e^{8 x} + C$ by $e^{x}$ .

$\frac{e^{x} y}{e^{x}} = \frac{\frac{1}{8} e^{8 x}}{e^{x}} + \frac{C}{e^{x}}$

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.

$\frac{e^{x} y}{e^{x}} = \frac{\frac{1}{8} e^{8 x}}{e^{x}} + \frac{C}{e^{x}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{8} e^{8 x}}{e^{x}} + \frac{C}{e^{x}}$

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 $e^{8 x}$ and $e^{x}$ .

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Step 7.3.1.1.1

Factor $e^{x}$ out of $\frac{1}{8} e^{8 x}$ .

$y = \frac{e^{x} (\frac{1}{8} e^{7 x})}{e^{x}} + \frac{C}{e^{x}}$

Step 7.3.1.1.2

Cancel the common factors.

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Step 7.3.1.1.2.1

Multiply by $1$ .

$y = \frac{e^{x} (\frac{1}{8} e^{7 x})}{e^{x} \cdot 1} + \frac{C}{e^{x}}$

Step 7.3.1.1.2.2

Cancel the common factor.

$y = \frac{e^{x} (\frac{1}{8} e^{7 x})}{e^{x} \cdot 1} + \frac{C}{e^{x}}$

Step 7.3.1.1.2.3

Rewrite the expression.

$y = \frac{\frac{1}{8} e^{7 x}}{1} + \frac{C}{e^{x}}$

Step 7.3.1.1.2.4

Divide $\frac{1}{8} e^{7 x}$ by $1$ .

$y = \frac{1}{8} e^{7 x} + \frac{C}{e^{x}}$

Step 7.3.1.2

Combine $\frac{1}{8}$ and $e^{7 x}$ .

$y = \frac{e^{7 x}}{8} + \frac{C}{e^{x}}$