Calculus Examples

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

Step 1

Subtract $y$ from both sides of the equation.

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = - 1$ .

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Step 2.1

Set up the integration.

$e^{\int - 1 d x}$

Step 2.2

Apply the constant rule.

$e^{- x + C}$

Step 2.3

Remove the constant of integration.

$e^{- x}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Rewrite using the commutative property of multiplication.

$e^{- x} \frac{d y}{d x} - e^{- x} y = e^{- x} e^{2 x} + e^{- x} \cdot - 1$

Step 3.3

Simplify each term.

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Step 3.3.1

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

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Step 3.3.1.1

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

$e^{- x} \frac{d y}{d x} - e^{- x} y = e^{- x + 2 x} + e^{- x} \cdot - 1$

Step 3.3.1.2

Add $- x$ and $2 x$ .

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

Step 3.3.2

Move $- 1$ to the left of $e^{- x}$ .

$e^{- x} \frac{d y}{d x} - e^{- x} y = e^{x} - 1 \cdot e^{- x}$

Step 3.3.3

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

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

Step 3.4

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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Step 7.4.1

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

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Step 7.4.1.1

Differentiate $- x$ .

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

Step 7.4.1.2

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 7.4.1.3

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

$- 1 \cdot 1$

Step 7.4.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 7.4.2

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

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

Step 7.5

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

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

Step 7.6

Simplify.

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Step 7.6.1

Multiply $- 1$ by $- 1$ .

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

Step 7.6.2

Multiply $\int e^{u} d u$ by $1$ .

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

Step 7.7

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

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

Step 7.8

Simplify.

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

Step 7.9

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

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

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Step 8.3.1.1.1

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

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

Step 8.3.1.1.2

Cancel the common factors.

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Step 8.3.1.1.2.1

Multiply by $1$ .

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

Step 8.3.1.1.2.2

Cancel the common factor.

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

Step 8.3.1.1.2.3

Rewrite the expression.

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

Step 8.3.1.1.2.4

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

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

Step 8.3.1.2

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

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Step 8.3.1.2.1

Cancel the common factor.

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

Step 8.3.1.2.2

Rewrite the expression.

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