Calculus Examples

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

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 $e^{- 3 x}$ .

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

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.

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

Step 2.3.2

Add $- 3 x$ and $2 x$ .

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

Step 2.4

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

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

Step 3

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

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

Step 4

Set up an integral on each side.

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

Step 5

Integrate the left side.

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

Step 6

Integrate the right side.

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Step 6.1

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

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Step 6.1.1

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

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Step 6.1.1.1

Differentiate $- x$ .

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

Step 6.1.1.2

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 6.1.1.3

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

$- 1 \cdot 1$

Step 6.1.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 6.1.2

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

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

Step 6.2

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

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

Step 6.3

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

$e^{- 3 x} y = - (e^{u} + C)$

Step 6.4

Simplify.

$e^{- 3 x} y = - e^{u} + C$

Step 6.5

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

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

Step 7

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

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Step 7.1

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

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

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.

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

Step 7.2.1.2

Divide $y$ by $1$ .

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

Step 7.3

Simplify the right side.

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Step 7.3.1

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

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Step 7.3.1.1

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

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

Step 7.3.1.2

Cancel the common factors.

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Step 7.3.1.2.1

Multiply by $1$ .

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

Step 7.3.1.2.2

Cancel the common factor.

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

Step 7.3.1.2.3

Rewrite the expression.

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

Step 7.3.1.2.4

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

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