Calculus Examples

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

Step 1

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

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Step 1.1

Set up the integration.

$e^{\int 2 d x}$

Step 1.2

Apply the constant rule.

$e^{2 x + C}$

Step 1.3

Remove the constant of integration.

$e^{2 x}$

Step 2

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

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Step 2.1

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

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

Multiply $e^{2 x}$ by $e^{x}$ by adding the exponents.

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Step 2.3.1

Move $e^{x}$ .

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

Step 2.3.2

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

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

Step 2.3.3

Add $x$ and $2 x$ .

$e^{2 x} \frac{d y}{d x} + 2 e^{2 x} y = e^{3 x} x$

Step 2.4

Reorder factors in $e^{2 x} \frac{d y}{d x} + 2 e^{2 x} y = e^{3 x} x$ .

$e^{2 x} \frac{d y}{d x} + 2 y e^{2 x} = x e^{3 x}$

Step 3

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

$\frac{d}{d x} [e^{2 x} y] = x e^{3 x}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{2 x} y] d x = \int x e^{3 x} d x$

Step 5

Integrate the left side.

$e^{2 x} y = \int x e^{3 x} d x$

Step 6

Integrate the right side.

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Step 6.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{3 x}$ .

$e^{2 x} y = x (\frac{1}{3} e^{3 x}) - \int \frac{1}{3} e^{3 x} d x$

Step 6.2

Simplify.

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Step 6.2.1

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

$e^{2 x} y = x \frac{e^{3 x}}{3} - \int \frac{1}{3} e^{3 x} d x$

Step 6.2.2

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

$e^{2 x} y = \frac{x e^{3 x}}{3} - \int \frac{1}{3} e^{3 x} d x$

Step 6.3

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

$e^{2 x} y = \frac{x e^{3 x}}{3} - (\frac{1}{3} \int e^{3 x} d x)$

Step 6.4

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 6.4.1

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

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Step 6.4.1.1

Differentiate $3 x$ .

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

Step 6.4.1.2

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 6.4.1.3

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 6.4.1.4

Multiply $3$ by $1$ .

$3$

Step 6.4.2

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

$e^{2 x} y = \frac{x e^{3 x}}{3} - \frac{1}{3} \int e^{u} \frac{1}{3} d u$

Step 6.5

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

$e^{2 x} y = \frac{x e^{3 x}}{3} - \frac{1}{3} \int \frac{e^{u}}{3} d u$

Step 6.6

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

$e^{2 x} y = \frac{x e^{3 x}}{3} - \frac{1}{3} (\frac{1}{3} \int e^{u} d u)$

Step 6.7

Simplify.

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Step 6.7.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

$e^{2 x} y = \frac{x e^{3 x}}{3} - \frac{1}{3 \cdot 3} \int e^{u} d u$

Step 6.7.2

Multiply $3$ by $3$ .

$e^{2 x} y = \frac{x e^{3 x}}{3} - \frac{1}{9} \int e^{u} d u$

Step 6.8

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

$e^{2 x} y = \frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u} + C)$

Step 6.9

Rewrite $\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u} + C)$ as $\frac{1}{3} x e^{3 x} - \frac{1}{9} e^{u} + C$ .

$e^{2 x} y = \frac{1}{3} x e^{3 x} - \frac{1}{9} e^{u} + C$

Step 6.10

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

$e^{2 x} y = \frac{1}{3} x e^{3 x} - \frac{1}{9} e^{3 x} + C$

Step 7

Divide each term in $e^{2 x} y = \frac{1}{3} x e^{3 x} - \frac{1}{9} e^{3 x} + C$ by $e^{2 x}$ and simplify.

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Step 7.1

Divide each term in $e^{2 x} y = \frac{1}{3} x e^{3 x} - \frac{1}{9} e^{3 x} + C$ by $e^{2 x}$ .

$\frac{e^{2 x} y}{e^{2 x}} = \frac{\frac{1}{3} x e^{3 x}}{e^{2 x}} + \frac{- \frac{1}{9} e^{3 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

$\frac{e^{2 x} y}{e^{2 x}} = \frac{\frac{1}{3} x e^{3 x}}{e^{2 x}} + \frac{- \frac{1}{9} e^{3 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{3} x e^{3 x}}{e^{2 x}} + \frac{- \frac{1}{9} e^{3 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

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 $e^{2 x}$ .

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Step 7.3.1.1.1

Factor $e^{2 x}$ out of $\frac{1}{3} x e^{3 x}$ .

$y = \frac{e^{2 x} (\frac{1}{3} x e^{x})}{e^{2 x}} + \frac{- \frac{1}{9} e^{3 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 7.3.1.1.2

Cancel the common factors.

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Step 7.3.1.1.2.1

Multiply by $1$ .

$y = \frac{e^{2 x} (\frac{1}{3} x e^{x})}{e^{2 x} \cdot 1} + \frac{- \frac{1}{9} e^{3 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 7.3.1.1.2.2

Cancel the common factor.

$y = \frac{e^{2 x} (\frac{1}{3} x e^{x})}{e^{2 x} \cdot 1} + \frac{- \frac{1}{9} e^{3 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 7.3.1.1.2.3

Rewrite the expression.

$y = \frac{\frac{1}{3} x e^{x}}{1} + \frac{- \frac{1}{9} e^{3 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 7.3.1.1.2.4

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

$y = \frac{1}{3} x e^{x} + \frac{- \frac{1}{9} e^{3 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 7.3.1.2

Multiply $\frac{1}{3} x e^{x}$ .

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Step 7.3.1.2.1

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

$y = \frac{x}{3} e^{x} + \frac{- \frac{1}{9} e^{3 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 7.3.1.2.2

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

$y = \frac{x e^{x}}{3} + \frac{- \frac{1}{9} e^{3 x}}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 7.3.1.3

Cancel the common factor of $e^{3 x}$ and $e^{2 x}$ .

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Step 7.3.1.3.1

Factor $e^{2 x}$ out of $- \frac{1}{9} e^{3 x}$ .

$y = \frac{x e^{x}}{3} + \frac{e^{2 x} (- \frac{1}{9} e^{x})}{e^{2 x}} + \frac{C}{e^{2 x}}$

Step 7.3.1.3.2

Cancel the common factors.

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Step 7.3.1.3.2.1

Multiply by $1$ .

$y = \frac{x e^{x}}{3} + \frac{e^{2 x} (- \frac{1}{9} e^{x})}{e^{2 x} \cdot 1} + \frac{C}{e^{2 x}}$

Step 7.3.1.3.2.2

Cancel the common factor.

$y = \frac{x e^{x}}{3} + \frac{e^{2 x} (- \frac{1}{9} e^{x})}{e^{2 x} \cdot 1} + \frac{C}{e^{2 x}}$

Step 7.3.1.3.2.3

Rewrite the expression.

$y = \frac{x e^{x}}{3} + \frac{- \frac{1}{9} e^{x}}{1} + \frac{C}{e^{2 x}}$

Step 7.3.1.3.2.4

Divide $- \frac{1}{9} e^{x}$ by $1$ .

$y = \frac{x e^{x}}{3} - \frac{1}{9} e^{x} + \frac{C}{e^{2 x}}$

Step 7.3.1.4

Combine $e^{x}$ and $\frac{1}{9}$ .

$y = \frac{x e^{x}}{3} - \frac{e^{x}}{9} + \frac{C}{e^{2 x}}$