Calculus Examples

$\frac{d y}{d x} - 2 y = 4 - x$

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 $e^{- 2 x}$ .

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

Simplify each term.

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Step 2.3.1

Move $4$ to the left of $e^{- 2 x}$ .

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

Step 2.3.2

Rewrite using the commutative property of multiplication.

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

Step 2.4

Reorder factors in $e^{- 2 x} \frac{d y}{d x} - 2 e^{- 2 x} y = 4 e^{- 2 x} - e^{- 2 x} x$ .

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

Step 3

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

$\frac{d}{d x} [e^{- 2 x} y] = 4 e^{- 2 x} - x e^{- 2 x}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{- 2 x} y] d x = \int 4 e^{- 2 x} - x e^{- 2 x} d x$

Step 5

Integrate the left side.

$e^{- 2 x} y = \int 4 e^{- 2 x} - x e^{- 2 x} d x$

Step 6

Integrate the right side.

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Step 6.1

Split the single integral into multiple integrals.

$e^{- 2 x} y = \int 4 e^{- 2 x} d x + \int - x e^{- 2 x} d x$

Step 6.2

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$e^{- 2 x} y = 4 \int e^{- 2 x} d x + \int - x e^{- 2 x} d x$

Step 6.3

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

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Step 6.3.1

Let $u_{1} = - 2 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 6.3.1.1

Differentiate $- 2 x$ .

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

Step 6.3.1.2

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 6.3.1.3

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 6.3.1.4

Multiply $- 2$ by $1$ .

$- 2$

Step 6.3.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$e^{- 2 x} y = 4 \int e^{u_{1}} \frac{1}{- 2} d u_{1} + \int - x e^{- 2 x} d x$

Step 6.4

Simplify.

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Step 6.4.1

Move the negative in front of the fraction.

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

Step 6.4.2

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

$e^{- 2 x} y = 4 \int - \frac{e^{u_{1}}}{2} d u_{1} + \int - x e^{- 2 x} d x$

Step 6.5

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

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

Step 6.6

Multiply $- 1$ by $4$ .

$e^{- 2 x} y = - 4 \int \frac{e^{u_{1}}}{2} d u_{1} + \int - x e^{- 2 x} d x$

Step 6.7

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

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

Step 6.8

Simplify.

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Step 6.8.1

Combine $\frac{1}{2}$ and $- 4$ .

$e^{- 2 x} y = \frac{- 4}{2} \int e^{u_{1}} d u_{1} + \int - x e^{- 2 x} d x$

Step 6.8.2

Cancel the common factor of $- 4$ and $2$ .

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Step 6.8.2.1

Factor $2$ out of $- 4$ .

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

Step 6.8.2.2

Cancel the common factors.

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Step 6.8.2.2.1

Factor $2$ out of $2$ .

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

Step 6.8.2.2.2

Cancel the common factor.

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

Step 6.8.2.2.3

Rewrite the expression.

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

Step 6.8.2.2.4

Divide $- 2$ by $1$ .

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

Step 6.9

The integral of $e^{u_{1}}$ with respect to $u_{1}$ is $e^{u_{1}}$ .

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

Step 6.10

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

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

Step 6.11

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

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

Step 6.12

Simplify.

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Step 6.12.1

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

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

Step 6.12.2

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

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

Step 6.12.3

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

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

Step 6.13

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

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

Step 6.14

Simplify.

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Step 6.14.1

Multiply $- 1$ by $- 1$ .

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

Step 6.14.2

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

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

Step 6.15

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

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

Step 6.16

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

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Step 6.16.1

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

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Step 6.16.1.1

Differentiate $- 2 x$ .

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

Step 6.16.1.2

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 6.16.1.3

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 6.16.1.4

Multiply $- 2$ by $1$ .

$- 2$

Step 6.16.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

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

Step 6.17

Simplify.

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Step 6.17.1

Move the negative in front of the fraction.

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

Step 6.17.2

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

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

Step 6.18

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

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

Step 6.19

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

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

Step 6.20

Simplify.

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Step 6.20.1

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

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

Step 6.20.2

Multiply $2$ by $2$ .

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

Step 6.21

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

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

Step 6.22

Simplify.

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Step 6.22.1

Simplify.

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

Step 6.22.2

Simplify.

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Step 6.22.2.1

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

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

Step 6.22.2.2

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

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

Step 6.22.2.3

Combine $e^{u_{2}}$ and $\frac{1}{4}$ .

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

Step 6.23

Substitute back in for each integration substitution variable.

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Step 6.23.1

Replace all occurrences of $u_{1}$ with $- 2 x$ .

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

Step 6.23.2

Replace all occurrences of $u_{2}$ with $- 2 x$ .

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

Step 6.24

Simplify.

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Step 6.24.1

Apply the distributive property.

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

Step 6.24.2

Multiply $- - \frac{e^{- 2 x} x}{2}$ .

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Step 6.24.2.1

Multiply $- 1$ by $- 1$ .

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

Step 6.24.2.2

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

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

Step 6.24.3

Multiply $- - \frac{e^{- 2 x}}{4}$ .

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Step 6.24.3.1

Multiply $- 1$ by $- 1$ .

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

Step 6.24.3.2

Multiply $\frac{e^{- 2 x}}{4}$ by $1$ .

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

Step 6.24.4

Reorder factors in $\frac{e^{- 2 x} x}{2} + \frac{e^{- 2 x}}{4}$ .

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

Step 6.25

Reorder terms.

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

Step 7

Divide each term in $e^{- 2 x} y = - 2 e^{- 2 x} + \frac{1}{2} x e^{- 2 x} + \frac{1}{4} e^{- 2 x} + C$ by $e^{- 2 x}$ and simplify.

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Step 7.1

Divide each term in $e^{- 2 x} y = - 2 e^{- 2 x} + \frac{1}{2} x e^{- 2 x} + \frac{1}{4} e^{- 2 x} + C$ by $e^{- 2 x}$ .

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

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.

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

Step 7.2.1.2

Divide $y$ by $1$ .

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

Step 7.3

Simplify the right side.

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Step 7.3.1

Combine the numerators over the common denominator.

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

Step 7.3.2

Simplify each term.

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Step 7.3.2.1

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

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

Step 7.3.2.2

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

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

Step 7.3.2.3

Combine $\frac{1}{4}$ and $e^{- 2 x}$ .

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

Step 7.3.3

To write $- 2 e^{- 2 x}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 7.3.4

Simplify terms.

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Step 7.3.4.1

Combine $- 2 e^{- 2 x}$ and $\frac{4}{4}$ .

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

Step 7.3.4.2

Combine the numerators over the common denominator.

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

Step 7.3.4.3

Simplify each term.

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Step 7.3.4.3.1

Simplify the numerator.

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Step 7.3.4.3.1.1

Factor $e^{- 2 x}$ out of $- 2 e^{- 2 x} \cdot 4 + e^{- 2 x}$ .

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Step 7.3.4.3.1.1.1

Factor $e^{- 2 x}$ out of $- 2 e^{- 2 x} \cdot 4$ .

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

Step 7.3.4.3.1.1.2

Multiply by $1$ .

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

Step 7.3.4.3.1.1.3

Factor $e^{- 2 x}$ out of $e^{- 2 x} (- 2 \cdot 4) + e^{- 2 x} \cdot 1$ .

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

Step 7.3.4.3.1.2

Multiply $- 2$ by $4$ .

$y = \frac{\frac{x e^{- 2 x}}{2} + \frac{e^{- 2 x} (- 8 + 1)}{4} + C}{e^{- 2 x}}$

Step 7.3.4.3.1.3

Add $- 8$ and $1$ .

$y = \frac{\frac{x e^{- 2 x}}{2} + \frac{e^{- 2 x} \cdot - 7}{4} + C}{e^{- 2 x}}$

Step 7.3.4.3.2

Move $- 7$ to the left of $e^{- 2 x}$ .

$y = \frac{\frac{x e^{- 2 x}}{2} + \frac{- 7 \cdot e^{- 2 x}}{4} + C}{e^{- 2 x}}$

Step 7.3.4.3.3

Move the negative in front of the fraction.

$y = \frac{\frac{x e^{- 2 x}}{2} - \frac{7 e^{- 2 x}}{4} + C}{e^{- 2 x}}$

Step 7.3.5

Simplify the numerator.

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Step 7.3.5.1

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

$y = \frac{\frac{x e^{- 2 x}}{2} \cdot \frac{2}{2} - \frac{7 e^{- 2 x}}{4} + C}{e^{- 2 x}}$

Step 7.3.5.2

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.3.5.2.1

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

$y = \frac{\frac{x e^{- 2 x} \cdot 2}{2 \cdot 2} - \frac{7 e^{- 2 x}}{4} + C}{e^{- 2 x}}$

Step 7.3.5.2.2

Multiply $2$ by $2$ .

$y = \frac{\frac{x e^{- 2 x} \cdot 2}{4} - \frac{7 e^{- 2 x}}{4} + C}{e^{- 2 x}}$

Step 7.3.5.3

Combine the numerators over the common denominator.

$y = \frac{\frac{x e^{- 2 x} \cdot 2 - 7 e^{- 2 x}}{4} + C}{e^{- 2 x}}$

Step 7.3.5.4

Simplify the numerator.

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Step 7.3.5.4.1

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

$y = \frac{\frac{2 \cdot (x e^{- 2 x}) - 7 e^{- 2 x}}{4} + C}{e^{- 2 x}}$

Step 7.3.5.4.2

Factor $e^{- 2 x}$ out of $2 x e^{- 2 x} - 7 e^{- 2 x}$ .

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Step 7.3.5.4.2.1

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

$y = \frac{\frac{e^{- 2 x} (2 x) - 7 e^{- 2 x}}{4} + C}{e^{- 2 x}}$

Step 7.3.5.4.2.2

Factor $e^{- 2 x}$ out of $- 7 e^{- 2 x}$ .

$y = \frac{\frac{e^{- 2 x} (2 x) + e^{- 2 x} \cdot - 7}{4} + C}{e^{- 2 x}}$

Step 7.3.5.4.2.3

Factor $e^{- 2 x}$ out of $e^{- 2 x} (2 x) + e^{- 2 x} \cdot - 7$ .

$y = \frac{\frac{e^{- 2 x} (2 x - 7)}{4} + C}{e^{- 2 x}}$

Step 7.3.5.5

To write $C$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{\frac{e^{- 2 x} (2 x - 7)}{4} + C \cdot \frac{4}{4}}{e^{- 2 x}}$

Step 7.3.5.6

Combine $C$ and $\frac{4}{4}$ .

$y = \frac{\frac{e^{- 2 x} (2 x - 7)}{4} + \frac{C \cdot 4}{4}}{e^{- 2 x}}$

Step 7.3.5.7

Combine the numerators over the common denominator.

$y = \frac{\frac{e^{- 2 x} (2 x - 7) + C \cdot 4}{4}}{e^{- 2 x}}$

Step 7.3.5.8

Simplify the numerator.

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Step 7.3.5.8.1

Apply the distributive property.

$y = \frac{\frac{e^{- 2 x} (2 x) + e^{- 2 x} \cdot - 7 + C \cdot 4}{4}}{e^{- 2 x}}$

Step 7.3.5.8.2

Rewrite using the commutative property of multiplication.

$y = \frac{\frac{2 e^{- 2 x} x + e^{- 2 x} \cdot - 7 + C \cdot 4}{4}}{e^{- 2 x}}$

Step 7.3.5.8.3

Move $- 7$ to the left of $e^{- 2 x}$ .

$y = \frac{\frac{2 e^{- 2 x} x - 7 \cdot e^{- 2 x} + C \cdot 4}{4}}{e^{- 2 x}}$

Step 7.3.5.8.4

Move $4$ to the left of $C$ .

$y = \frac{\frac{2 e^{- 2 x} x - 7 e^{- 2 x} + 4 C}{4}}{e^{- 2 x}}$

Step 7.3.6

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{2 e^{- 2 x} x - 7 e^{- 2 x} + 4 C}{4} \cdot \frac{1}{e^{- 2 x}}$

Step 7.3.7

Multiply $\frac{2 e^{- 2 x} x - 7 e^{- 2 x} + 4 C}{4}$ by $\frac{1}{e^{- 2 x}}$ .

$y = \frac{2 e^{- 2 x} x - 7 e^{- 2 x} + 4 C}{4 e^{- 2 x}}$

Step 7.3.8

Reorder factors in $\frac{2 e^{- 2 x} x - 7 e^{- 2 x} + 4 C}{4 e^{- 2 x}}$ .

$y = \frac{2 x e^{- 2 x} - 7 e^{- 2 x} + 4 C}{4 e^{- 2 x}}$