Calculus Examples

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

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 3$ .

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Step 1.1

Set up the integration.

$e^{\int 3 d x}$

Step 1.2

Apply the constant rule.

$e^{3 x + C}$

Step 1.3

Remove the constant of integration.

$e^{3 x}$

Step 2

Multiply each term by the integrating factor $e^{3 x}$ .

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Step 2.1

Multiply each term by $e^{3 x}$ .

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

Multiply $e^{3 x}$ by $e^{- 2 x}$ 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.

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

Step 2.3.2

Subtract $2 x$ from $3 x$ .

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

Step 2.4

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

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

Step 3

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

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

Step 4

Set up an integral on each side.

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

Step 5

Integrate the left side.

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

Step 6

Integrate the right side.

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Step 6.1

Split the single integral into multiple integrals.

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

Step 6.2

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

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

Step 6.3

Simplify.

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Step 6.3.1

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

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

Step 6.3.2

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

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

Step 6.4

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

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

Step 6.5

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 6.5.1

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

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Step 6.5.1.1

Differentiate $3 x$ .

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

Step 6.5.1.2

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 6.5.1.3

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 6.5.1.4

Multiply $3$ by $1$ .

$3$

Step 6.5.2

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

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

Step 6.6

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

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

Step 6.7

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

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

Step 6.8

Simplify.

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Step 6.8.1

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

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

Step 6.8.2

Multiply $3$ by $3$ .

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

Step 6.9

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

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

Step 6.10

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

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

Step 6.11

Simplify.

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

Step 6.12

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $y$ by $1$ .

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

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 $e^{3 x}$ .

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Step 7.3.1.1.1

Cancel the common factor.

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

Step 7.3.1.1.2

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

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

Step 7.3.1.2

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

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Step 7.3.1.2.1

Cancel the common factor.

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

Step 7.3.1.2.2

Divide $- \frac{1}{9}$ by $1$ .

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

Step 7.3.1.3

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

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Step 7.3.1.3.1

Factor $e^{3 x}$ out of $e^{x}$ .

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

Step 7.3.1.3.2

Cancel the common factors.

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Step 7.3.1.3.2.1

Multiply by $1$ .

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

Step 7.3.1.3.2.2

Cancel the common factor.

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

Step 7.3.1.3.2.3

Rewrite the expression.

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

Step 7.3.1.3.2.4

Divide $e^{- 2 x}$ by $1$ .

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

Step 7.3.2

Subtract $\frac{1}{9}$ from $\frac{1}{3} x$ .

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Step 7.3.2.1

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

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

Step 7.3.2.2

To write $x \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

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

Step 7.3.2.3

Combine $x \frac{1}{3}$ and $\frac{9}{9}$ .

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

Step 7.3.2.4

Combine the numerators over the common denominator.

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

Step 7.3.3

Simplify the numerator.

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Step 7.3.3.1

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

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

Step 7.3.3.2

Cancel the common factor of $3$ .

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Step 7.3.3.2.1

Factor $3$ out of $9$ .

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

Step 7.3.3.2.2

Cancel the common factor.

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

Step 7.3.3.2.3

Rewrite the expression.

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

Step 7.3.3.3

Move $3$ to the left of $x$ .

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

Step 7.3.4

To write $e^{- 2 x}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

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

Step 7.3.5

Simplify terms.

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Step 7.3.5.1

Combine $e^{- 2 x}$ and $\frac{9}{9}$ .

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

Step 7.3.5.2

Combine the numerators over the common denominator.

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

Step 7.3.6

Simplify the numerator.

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Step 7.3.6.1

Move $9$ to the left of $e^{- 2 x}$ .

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

Step 7.3.6.2

Reorder terms.

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

Step 7.3.7

To write $\frac{3 x + 9 e^{- 2 x} - 1}{9}$ as a fraction with a common denominator, multiply by $\frac{e^{3 x}}{e^{3 x}}$ .

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

Step 7.3.8

To write $\frac{C}{e^{3 x}}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

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

Step 7.3.9

Write each expression with a common denominator of $9 e^{3 x}$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.3.9.1

Multiply $\frac{3 x + 9 e^{- 2 x} - 1}{9}$ by $\frac{e^{3 x}}{e^{3 x}}$ .

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

Step 7.3.9.2

Multiply $\frac{C}{e^{3 x}}$ by $\frac{9}{9}$ .

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

Step 7.3.9.3

Reorder the factors of $e^{3 x} \cdot 9$ .

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

Step 7.3.10

Combine the numerators over the common denominator.

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

Step 7.3.11

Simplify the numerator.

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Step 7.3.11.1

Apply the distributive property.

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

Step 7.3.11.2

Simplify.

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Step 7.3.11.2.1

Multiply $e^{- 2 x}$ by $e^{3 x}$ by adding the exponents.

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Step 7.3.11.2.1.1

Move $e^{3 x}$ .

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

Step 7.3.11.2.1.2

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

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

Step 7.3.11.2.1.3

Subtract $2 x$ from $3 x$ .

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

Step 7.3.11.2.2

Rewrite $- 1 e^{3 x}$ as $- e^{3 x}$ .

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

Step 7.3.11.3

Move $9$ to the left of $C$ .

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