Calculus Examples

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

Step 1

Subtract $2 y$ from both sides of the equation.

$\frac{d y}{d x} - 2 y = 2 x$

Step 2

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

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Step 2.1

Set up the integration.

$e^{\int - 2 d x}$

Step 2.2

Apply the constant rule.

$e^{- 2 x + C}$

Step 2.3

Remove the constant of integration.

$e^{- 2 x}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 7.2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{- 2 x}$ .

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

Step 7.3

Simplify.

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Step 7.3.1

Combine $e^{- 2 x}$ and $\frac{1}{2}$ .

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

Step 7.3.2

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

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

Step 7.3.3

Combine $e^{- 2 x}$ and $\frac{1}{2}$ .

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

Step 7.4

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

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

Step 7.5

Simplify.

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Step 7.5.1

Multiply $- 1$ by $- 1$ .

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

Step 7.5.2

Multiply $\int \frac{e^{- 2 x}}{2} d x$ by $1$ .

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

Step 7.6

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 7.7

Let $u = - 2 x$ . Then $d u = - 2 d x$ , so $- \frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.7.1

Let $u = - 2 x$ . Find $\frac{d u}{d x}$ .

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Step 7.7.1.1

Differentiate $- 2 x$ .

$\frac{d}{d x} [- 2 x]$

Step 7.7.1.2

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$- 2 \frac{d}{d x} [x]$

Step 7.7.1.3

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

$- 2 \cdot 1$

Step 7.7.1.4

Multiply $- 2$ by $1$ .

$- 2$

Step 7.7.2

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

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

Step 7.8

Simplify.

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Step 7.8.1

Move the negative in front of the fraction.

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

Step 7.8.2

Combine $e^{u}$ and $\frac{1}{2}$ .

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

Step 7.9

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

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

Step 7.10

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 7.11

Simplify.

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Step 7.11.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

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

Step 7.11.2

Multiply $2$ by $2$ .

$e^{- 2 x} y = 2 (- \frac{x e^{- 2 x}}{2} - \frac{1}{4} \int e^{u} d u)$

Step 7.12

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

$e^{- 2 x} y = 2 (- \frac{x e^{- 2 x}}{2} - \frac{1}{4} (e^{u} + C))$

Step 7.13

Simplify.

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Step 7.13.1

Rewrite $2 (- \frac{x e^{- 2 x}}{2} - \frac{1}{4} (e^{u} + C))$ as $2 (- \frac{1}{2} x e^{- 2 x} - \frac{1}{4} e^{u}) + C$ .

$e^{- 2 x} y = 2 (- \frac{1}{2} x e^{- 2 x} - \frac{1}{4} e^{u}) + C$

Step 7.13.2

Simplify.

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Step 7.13.2.1

Combine $x$ and $\frac{1}{2}$ .

$e^{- 2 x} y = 2 (- \frac{x}{2} e^{- 2 x} - \frac{1}{4} e^{u}) + C$

Step 7.13.2.2

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

$e^{- 2 x} y = 2 (- \frac{e^{- 2 x} x}{2} - \frac{1}{4} e^{u}) + C$

Step 7.13.2.3

Combine $e^{u}$ and $\frac{1}{4}$ .

$e^{- 2 x} y = 2 (- \frac{e^{- 2 x} x}{2} - \frac{e^{u}}{4}) + C$

Step 7.14

Replace all occurrences of $u$ with $- 2 x$ .

$e^{- 2 x} y = 2 (- \frac{e^{- 2 x} x}{2} - \frac{e^{- 2 x}}{4}) + C$

Step 7.15

Simplify.

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Step 7.15.1

Apply the distributive property.

$e^{- 2 x} y = 2 (- \frac{e^{- 2 x} x}{2}) + 2 (- \frac{e^{- 2 x}}{4}) + C$

Step 7.15.2

Cancel the common factor of $2$ .

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Step 7.15.2.1

Move the leading negative in $- \frac{e^{- 2 x} x}{2}$ into the numerator.

$e^{- 2 x} y = 2 \frac{- e^{- 2 x} x}{2} + 2 (- \frac{e^{- 2 x}}{4}) + C$

Step 7.15.2.2

Cancel the common factor.

$e^{- 2 x} y = 2 \frac{- e^{- 2 x} x}{2} + 2 (- \frac{e^{- 2 x}}{4}) + C$

Step 7.15.2.3

Rewrite the expression.

$e^{- 2 x} y = - e^{- 2 x} x + 2 (- \frac{e^{- 2 x}}{4}) + C$

Step 7.15.3

Cancel the common factor of $2$ .

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Step 7.15.3.1

Move the leading negative in $- \frac{e^{- 2 x}}{4}$ into the numerator.

$e^{- 2 x} y = - e^{- 2 x} x + 2 \frac{- e^{- 2 x}}{4} + C$

Step 7.15.3.2

Factor $2$ out of $4$ .

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

Step 7.15.3.3

Cancel the common factor.

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

Step 7.15.3.4

Rewrite the expression.

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

Step 7.15.4

Move the negative in front of the fraction.

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

Step 7.15.5

Combine $- e^{- 2 x} x$ and $- \frac{e^{- 2 x}}{2}$ using a common denominator.

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Step 7.15.5.1

Move $e^{- 2 x}$ .

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

Step 7.15.5.2

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

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

Step 7.15.5.3

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

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

Step 7.15.5.4

Combine the numerators over the common denominator.

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

Step 7.15.6

Simplify the numerator.

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Step 7.15.6.1

Factor $e^{- 2 x}$ out of $- x e^{- 2 x} \cdot 2 - e^{- 2 x}$ .

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Step 7.15.6.1.1

Factor $e^{- 2 x}$ out of $- x e^{- 2 x} \cdot 2$ .

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

Step 7.15.6.1.2

Factor $e^{- 2 x}$ out of $- e^{- 2 x}$ .

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

Step 7.15.6.1.3

Factor $e^{- 2 x}$ out of $e^{- 2 x} (- x \cdot 2) + e^{- 2 x} \cdot - 1$ .

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

Step 7.15.6.2

Multiply $2$ by $- 1$ .

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

Step 7.15.7

Factor $- 1$ out of $- 2 x$ .

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

Step 7.15.8

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 7.15.9

Factor $- 1$ out of $- (2 x) - 1 (1)$ .

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

Step 7.15.10

Rewrite $- (2 x + 1)$ as $- 1 (2 x + 1)$ .

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

Step 7.15.11

Move the negative in front of the fraction.

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

Step 7.16

Remove parentheses.

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

Step 8

Solve for $y$ .

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Step 8.1

Combine $e^{- 2 x}$ and $\frac{1}{2}$ .

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

Step 8.2

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

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Step 8.2.1

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

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

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

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

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Step 8.2.2.1.1

Cancel the common factor.

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

Step 8.2.2.1.2

Divide $y$ by $1$ .

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

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Combine the numerators over the common denominator.

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

Step 8.2.3.2

Simplify each term.

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Step 8.2.3.2.1

Apply the distributive property.

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

Step 8.2.3.2.2

Cancel the common factor of $2$ .

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Step 8.2.3.2.2.1

Move the leading negative in $- \frac{e^{- 2 x}}{2}$ into the numerator.

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

Step 8.2.3.2.2.2

Factor $2$ out of $2 x$ .

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

Step 8.2.3.2.2.3

Cancel the common factor.

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

Step 8.2.3.2.2.4

Rewrite the expression.

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

Step 8.2.3.2.3

Multiply $- 1$ by $1$ .

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

Step 8.2.3.2.4

Combine $- e^{- 2 x} x$ and $- \frac{e^{- 2 x}}{2}$ using a common denominator.

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Step 8.2.3.2.4.1

Move $e^{- 2 x}$ .

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

Step 8.2.3.2.4.2

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

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

Step 8.2.3.2.4.3

Combine $- 1 x e^{- 2 x}$ and $\frac{2}{2}$ .

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

Step 8.2.3.2.4.4

Combine the numerators over the common denominator.

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

Step 8.2.3.2.5

Simplify the numerator.

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Step 8.2.3.2.5.1

Factor $e^{- 2 x}$ out of $- 1 x e^{- 2 x} \cdot 2 - e^{- 2 x}$ .

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Step 8.2.3.2.5.1.1

Factor $e^{- 2 x}$ out of $- 1 x e^{- 2 x} \cdot 2$ .

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

Step 8.2.3.2.5.1.2

Factor $e^{- 2 x}$ out of $- e^{- 2 x}$ .

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

Step 8.2.3.2.5.1.3

Factor $e^{- 2 x}$ out of $e^{- 2 x} (- 1 x \cdot 2) + e^{- 2 x} \cdot - 1$ .

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

Step 8.2.3.2.5.2

Rewrite $- 1 x$ as $- x$ .

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

Step 8.2.3.2.5.3

Multiply $2$ by $- 1$ .

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

Step 8.2.3.3

To write $C$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 8.2.3.4

Simplify terms.

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Step 8.2.3.4.1

Combine $C$ and $\frac{2}{2}$ .

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

Step 8.2.3.4.2

Combine the numerators over the common denominator.

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

Step 8.2.3.5

Simplify the numerator.

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Step 8.2.3.5.1

Apply the distributive property.

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

Step 8.2.3.5.2

Rewrite using the commutative property of multiplication.

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

Step 8.2.3.5.3

Move $- 1$ to the left of $e^{- 2 x}$ .

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

Step 8.2.3.5.4

Rewrite $- 1 e^{- 2 x}$ as $- e^{- 2 x}$ .

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

Step 8.2.3.5.5

Move $2$ to the left of $C$ .

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

Step 8.2.3.6

Simplify with factoring out.

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Step 8.2.3.6.1

Factor $- 1$ out of $- 2 e^{- 2 x} x$ .

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

Step 8.2.3.6.2

Factor $- 1$ out of $- e^{- 2 x}$ .

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

Step 8.2.3.6.3

Factor $- 1$ out of $- (2 e^{- 2 x} x) - (e^{- 2 x})$ .

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

Step 8.2.3.6.4

Factor $- 1$ out of $2 C$ .

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

Step 8.2.3.6.5

Factor $- 1$ out of $- (2 e^{- 2 x} x + e^{- 2 x}) - (- 2 C)$ .

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

Step 8.2.3.6.6

Simplify the expression.

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Step 8.2.3.6.6.1

Rewrite $- (2 e^{- 2 x} x + e^{- 2 x} - 2 C)$ as $- 1 (2 e^{- 2 x} x + e^{- 2 x} - 2 C)$ .

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

Step 8.2.3.6.6.2

Move the negative in front of the fraction.

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

Step 8.2.3.6.6.3

Reorder factors in $- \frac{2 e^{- 2 x} x + e^{- 2 x} - 2 C}{2}$ .

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

Step 8.2.3.7

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2.3.8

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

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

Step 8.2.3.9

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

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