Calculus Examples

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

Step 1

Subtract $4 y$ from both sides of the equation.

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = - 4$ .

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Step 2.1

Set up the integration.

$e^{\int - 4 d x}$

Step 2.2

Apply the constant rule.

$e^{- 4 x + C}$

Step 2.3

Remove the constant of integration.

$e^{- 4 x}$

Step 3

Multiply each term by the integrating factor $e^{- 4 x}$ .

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Step 3.1

Multiply each term by $e^{- 4 x}$ .

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

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

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

Step 3.4

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

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

Step 7.2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{- 4 x}$ .

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

Step 7.3

Simplify.

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Step 7.3.1

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

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

Step 7.3.2

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

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

Step 7.4

Since $- \frac{1}{4}$ is constant with respect to $x$ , move $- \frac{1}{4}$ out of the integral.

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

Step 7.5

Simplify.

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Step 7.5.1

Multiply $- 1$ by $- 1$ .

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

Step 7.5.2

Multiply $\frac{1}{4}$ by $1$ .

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

Step 7.6

Let $u_{1} = - 4 x$ . Then $d u_{1} = - 4 d x$ , so $- \frac{1}{4} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 7.6.1

Let $u_{1} = - 4 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 7.6.1.1

Differentiate $- 4 x$ .

$\frac{d}{d x} [- 4 x]$

Step 7.6.1.2

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$- 4 \frac{d}{d x} [x]$

Step 7.6.1.3

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

$- 4 \cdot 1$

Step 7.6.1.4

Multiply $- 4$ by $1$ .

$- 4$

Step 7.6.2

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

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

Step 7.7

Simplify.

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Step 7.7.1

Move the negative in front of the fraction.

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

Step 7.7.2

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

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

Step 7.8

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

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

Step 7.9

Since $\frac{1}{4}$ is constant with respect to $u_{1}$ , move $\frac{1}{4}$ out of the integral.

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

Step 7.10

Simplify.

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Step 7.10.1

Multiply $\frac{1}{4}$ by $\frac{1}{4}$ .

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

Step 7.10.2

Multiply $4$ by $4$ .

$e^{- 4 x} y = - \frac{x e^{- 4 x}}{4} - \frac{1}{16} \int e^{u_{1}} d u_{1} + \int - 2 e^{- 4 x} d x$

Step 7.11

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

$e^{- 4 x} y = - \frac{x e^{- 4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) + \int - 2 e^{- 4 x} d x$

Step 7.12

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

$e^{- 4 x} y = - \frac{x e^{- 4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 \int e^{- 4 x} d x$

Step 7.13

Let $u_{2} = - 4 x$ . Then $d u_{2} = - 4 d x$ , so $- \frac{1}{4} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 7.13.1

Let $u_{2} = - 4 x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 7.13.1.1

Differentiate $- 4 x$ .

$\frac{d}{d x} [- 4 x]$

Step 7.13.1.2

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$- 4 \frac{d}{d x} [x]$

Step 7.13.1.3

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

$- 4 \cdot 1$

Step 7.13.1.4

Multiply $- 4$ by $1$ .

$- 4$

Step 7.13.2

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

$e^{- 4 x} y = - \frac{x e^{- 4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 \int e^{u_{2}} \frac{1}{- 4} d u_{2}$

Step 7.14

Simplify.

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Step 7.14.1

Move the negative in front of the fraction.

$e^{- 4 x} y = - \frac{x e^{- 4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 \int e^{u_{2}} (- \frac{1}{4}) d u_{2}$

Step 7.14.2

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

$e^{- 4 x} y = - \frac{x e^{- 4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 \int - \frac{e^{u_{2}}}{4} d u_{2}$

Step 7.15

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

$e^{- 4 x} y = - \frac{x e^{- 4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (- \int \frac{e^{u_{2}}}{4} d u_{2})$

Step 7.16

Multiply $- 1$ by $- 2$ .

$e^{- 4 x} y = - \frac{x e^{- 4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) + 2 \int \frac{e^{u_{2}}}{4} d u_{2}$

Step 7.17

Since $\frac{1}{4}$ is constant with respect to $u_{2}$ , move $\frac{1}{4}$ out of the integral.

$e^{- 4 x} y = - \frac{x e^{- 4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) + 2 (\frac{1}{4} \int e^{u_{2}} d u_{2})$

Step 7.18

Simplify.

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Step 7.18.1

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

$e^{- 4 x} y = - \frac{x e^{- 4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) + \frac{2}{4} \int e^{u_{2}} d u_{2}$

Step 7.18.2

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

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Step 7.18.2.1

Factor $2$ out of $2$ .

$e^{- 4 x} y = - \frac{x e^{- 4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) + \frac{2 (1)}{4} \int e^{u_{2}} d u_{2}$

Step 7.18.2.2

Cancel the common factors.

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Step 7.18.2.2.1

Factor $2$ out of $4$ .

$e^{- 4 x} y = - \frac{x e^{- 4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) + \frac{2 \cdot 1}{2 \cdot 2} \int e^{u_{2}} d u_{2}$

Step 7.18.2.2.2

Cancel the common factor.

$e^{- 4 x} y = - \frac{x e^{- 4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) + \frac{2 \cdot 1}{2 \cdot 2} \int e^{u_{2}} d u_{2}$

Step 7.18.2.2.3

Rewrite the expression.

$e^{- 4 x} y = - \frac{x e^{- 4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) + \frac{1}{2} \int e^{u_{2}} d u_{2}$

Step 7.19

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

$e^{- 4 x} y = - \frac{x e^{- 4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) + \frac{1}{2} (e^{u_{2}} + C)$

Step 7.20

Simplify.

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Step 7.20.1

Simplify.

$e^{- 4 x} y = - \frac{1}{4} x e^{- 4 x} - \frac{1}{16} e^{u_{1}} + \frac{1}{2} e^{u_{2}} + C$

Step 7.20.2

Simplify.

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Step 7.20.2.1

Combine $x$ and $\frac{1}{4}$ .

$e^{- 4 x} y = - \frac{x}{4} e^{- 4 x} - \frac{1}{16} e^{u_{1}} + \frac{1}{2} e^{u_{2}} + C$

Step 7.20.2.2

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

$e^{- 4 x} y = - \frac{e^{- 4 x} x}{4} - \frac{1}{16} e^{u_{1}} + \frac{1}{2} e^{u_{2}} + C$

Step 7.21

Substitute back in for each integration substitution variable.

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Step 7.21.1

Replace all occurrences of $u_{1}$ with $- 4 x$ .

$e^{- 4 x} y = - \frac{e^{- 4 x} x}{4} - \frac{1}{16} e^{- 4 x} + \frac{1}{2} e^{u_{2}} + C$

Step 7.21.2

Replace all occurrences of $u_{2}$ with $- 4 x$ .

$e^{- 4 x} y = - \frac{e^{- 4 x} x}{4} - \frac{1}{16} e^{- 4 x} + \frac{1}{2} e^{- 4 x} + C$

Step 7.22

Combine $e^{- 4 x}$ and $\frac{1}{16}$ .

$e^{- 4 x} y = - \frac{e^{- 4 x} x}{4} - \frac{e^{- 4 x}}{16} + \frac{1}{2} e^{- 4 x} + C$

Step 7.23

Reorder terms.

$e^{- 4 x} y = - \frac{1}{4} e^{- 4 x} x - \frac{1}{16} e^{- 4 x} + \frac{1}{2} e^{- 4 x} + C$

Step 8

Solve for $y$ .

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Step 8.1

Simplify.

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Step 8.1.1

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

$e^{- 4 x} y = - \frac{e^{- 4 x}}{4} x - \frac{1}{16} e^{- 4 x} + \frac{1}{2} e^{- 4 x} + C$

Step 8.1.2

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

$e^{- 4 x} y = - \frac{x e^{- 4 x}}{4} - \frac{1}{16} e^{- 4 x} + \frac{1}{2} e^{- 4 x} + C$

Step 8.1.3

Combine $e^{- 4 x}$ and $\frac{1}{16}$ .

$e^{- 4 x} y = - \frac{x e^{- 4 x}}{4} - \frac{e^{- 4 x}}{16} + \frac{1}{2} e^{- 4 x} + C$

Step 8.2

Divide each term in $e^{- 4 x} y = - \frac{x e^{- 4 x}}{4} - \frac{e^{- 4 x}}{16} + \frac{1}{2} \cdot e^{- 4 x} + C$ by $e^{- 4 x}$ and simplify.

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Step 8.2.1

Divide each term in $e^{- 4 x} y = - \frac{x e^{- 4 x}}{4} - \frac{e^{- 4 x}}{16} + \frac{1}{2} \cdot e^{- 4 x} + C$ by $e^{- 4 x}$ .

$\frac{e^{- 4 x} y}{e^{- 4 x}} = \frac{- \frac{x e^{- 4 x}}{4}}{e^{- 4 x}} + \frac{- \frac{e^{- 4 x}}{16}}{e^{- 4 x}} + \frac{\frac{1}{2} \cdot e^{- 4 x}}{e^{- 4 x}} + \frac{C}{e^{- 4 x}}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $e^{- 4 x}$ .

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{e^{- 4 x} y}{e^{- 4 x}} = \frac{- \frac{x e^{- 4 x}}{4}}{e^{- 4 x}} + \frac{- \frac{e^{- 4 x}}{16}}{e^{- 4 x}} + \frac{\frac{1}{2} \cdot e^{- 4 x}}{e^{- 4 x}} + \frac{C}{e^{- 4 x}}$

Step 8.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{x e^{- 4 x}}{4}}{e^{- 4 x}} + \frac{- \frac{e^{- 4 x}}{16}}{e^{- 4 x}} + \frac{\frac{1}{2} \cdot e^{- 4 x}}{e^{- 4 x}} + \frac{C}{e^{- 4 x}}$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Combine the numerators over the common denominator.

$y = \frac{- \frac{x e^{- 4 x}}{4} - \frac{e^{- 4 x}}{16} + \frac{1}{2} \cdot e^{- 4 x} + C}{e^{- 4 x}}$

Step 8.2.3.2

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

$y = \frac{- \frac{x e^{- 4 x}}{4} - \frac{e^{- 4 x}}{16} + \frac{e^{- 4 x}}{2} + C}{e^{- 4 x}}$

Step 8.2.3.3

To write $\frac{e^{- 4 x}}{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = \frac{- \frac{x e^{- 4 x}}{4} - \frac{e^{- 4 x}}{16} + \frac{e^{- 4 x}}{2} \cdot \frac{8}{8} + C}{e^{- 4 x}}$

Step 8.2.3.4

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.2.3.4.1

Multiply $\frac{e^{- 4 x}}{2}$ by $\frac{8}{8}$ .

$y = \frac{- \frac{x e^{- 4 x}}{4} - \frac{e^{- 4 x}}{16} + \frac{e^{- 4 x} \cdot 8}{2 \cdot 8} + C}{e^{- 4 x}}$

Step 8.2.3.4.2

Multiply $2$ by $8$ .

$y = \frac{- \frac{x e^{- 4 x}}{4} - \frac{e^{- 4 x}}{16} + \frac{e^{- 4 x} \cdot 8}{16} + C}{e^{- 4 x}}$

Step 8.2.3.5

Combine the numerators over the common denominator.

$y = \frac{- \frac{x e^{- 4 x}}{4} + \frac{- e^{- 4 x} + e^{- 4 x} \cdot 8}{16} + C}{e^{- 4 x}}$

Step 8.2.3.6

Add $- e^{- 4 x}$ and $e^{- 4 x} \cdot 8$ .

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Step 8.2.3.6.1

Reorder $e^{- 4 x}$ and $8$ .

$y = \frac{- \frac{x e^{- 4 x}}{4} + \frac{- e^{- 4 x} + 8 \cdot e^{- 4 x}}{16} + C}{e^{- 4 x}}$

Step 8.2.3.6.2

Add $- e^{- 4 x}$ and $8 \cdot e^{- 4 x}$ .

$y = \frac{- \frac{x e^{- 4 x}}{4} + \frac{7 e^{- 4 x}}{16} + C}{e^{- 4 x}}$

Step 8.2.3.7

Simplify the numerator.

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Step 8.2.3.7.1

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

$y = \frac{- \frac{x e^{- 4 x}}{4} \cdot \frac{4}{4} + \frac{7 e^{- 4 x}}{16} + C}{e^{- 4 x}}$

Step 8.2.3.7.2

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.2.3.7.2.1

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

$y = \frac{- \frac{x e^{- 4 x} \cdot 4}{4 \cdot 4} + \frac{7 e^{- 4 x}}{16} + C}{e^{- 4 x}}$

Step 8.2.3.7.2.2

Multiply $4$ by $4$ .

$y = \frac{- \frac{x e^{- 4 x} \cdot 4}{16} + \frac{7 e^{- 4 x}}{16} + C}{e^{- 4 x}}$

Step 8.2.3.7.3

Combine the numerators over the common denominator.

$y = \frac{\frac{- x e^{- 4 x} \cdot 4 + 7 e^{- 4 x}}{16} + C}{e^{- 4 x}}$

Step 8.2.3.7.4

Simplify the numerator.

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Step 8.2.3.7.4.1

Factor $e^{- 4 x}$ out of $- x e^{- 4 x} \cdot 4 + 7 e^{- 4 x}$ .

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Step 8.2.3.7.4.1.1

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

$y = \frac{\frac{e^{- 4 x} (- x \cdot 4) + 7 e^{- 4 x}}{16} + C}{e^{- 4 x}}$

Step 8.2.3.7.4.1.2

Factor $e^{- 4 x}$ out of $7 e^{- 4 x}$ .

$y = \frac{\frac{e^{- 4 x} (- x \cdot 4) + e^{- 4 x} \cdot 7}{16} + C}{e^{- 4 x}}$

Step 8.2.3.7.4.1.3

Factor $e^{- 4 x}$ out of $e^{- 4 x} (- x \cdot 4) + e^{- 4 x} \cdot 7$ .

$y = \frac{\frac{e^{- 4 x} (- x \cdot 4 + 7)}{16} + C}{e^{- 4 x}}$

Step 8.2.3.7.4.2

Multiply $4$ by $- 1$ .

$y = \frac{\frac{e^{- 4 x} (- 4 x + 7)}{16} + C}{e^{- 4 x}}$

Step 8.2.3.7.5

To write $C$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$y = \frac{\frac{e^{- 4 x} (- 4 x + 7)}{16} + C \cdot \frac{16}{16}}{e^{- 4 x}}$

Step 8.2.3.7.6

Combine $C$ and $\frac{16}{16}$ .

$y = \frac{\frac{e^{- 4 x} (- 4 x + 7)}{16} + \frac{C \cdot 16}{16}}{e^{- 4 x}}$

Step 8.2.3.7.7

Combine the numerators over the common denominator.

$y = \frac{\frac{e^{- 4 x} (- 4 x + 7) + C \cdot 16}{16}}{e^{- 4 x}}$

Step 8.2.3.7.8

Simplify the numerator.

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Step 8.2.3.7.8.1

Apply the distributive property.

$y = \frac{\frac{e^{- 4 x} (- 4 x) + e^{- 4 x} \cdot 7 + C \cdot 16}{16}}{e^{- 4 x}}$

Step 8.2.3.7.8.2

Rewrite using the commutative property of multiplication.

$y = \frac{\frac{- 4 e^{- 4 x} x + e^{- 4 x} \cdot 7 + C \cdot 16}{16}}{e^{- 4 x}}$

Step 8.2.3.7.8.3

Move $7$ to the left of $e^{- 4 x}$ .

$y = \frac{\frac{- 4 e^{- 4 x} x + 7 \cdot e^{- 4 x} + C \cdot 16}{16}}{e^{- 4 x}}$

Step 8.2.3.7.8.4

Move $16$ to the left of $C$ .

$y = \frac{\frac{- 4 e^{- 4 x} x + 7 e^{- 4 x} + 16 C}{16}}{e^{- 4 x}}$

Step 8.2.3.8

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{- 4 e^{- 4 x} x + 7 e^{- 4 x} + 16 C}{16} \cdot \frac{1}{e^{- 4 x}}$

Step 8.2.3.9

Multiply $\frac{- 4 e^{- 4 x} x + 7 e^{- 4 x} + 16 C}{16}$ by $\frac{1}{e^{- 4 x}}$ .

$y = \frac{- 4 e^{- 4 x} x + 7 e^{- 4 x} + 16 C}{16 e^{- 4 x}}$

Step 8.2.3.10

Factor $- 1$ out of $- 4 e^{- 4 x} x$ .

$y = \frac{- (4 e^{- 4 x} x) + 7 e^{- 4 x} + 16 C}{16 e^{- 4 x}}$

Step 8.2.3.11

Factor $- 1$ out of $7 e^{- 4 x}$ .

$y = \frac{- (4 e^{- 4 x} x) - (- 7 e^{- 4 x}) + 16 C}{16 e^{- 4 x}}$

Step 8.2.3.12

Factor $- 1$ out of $- (4 e^{- 4 x} x) - (- 7 e^{- 4 x})$ .

$y = \frac{- (4 e^{- 4 x} x - 7 e^{- 4 x}) + 16 C}{16 e^{- 4 x}}$

Step 8.2.3.13

Factor $- 1$ out of $16 C$ .

$y = \frac{- (4 e^{- 4 x} x - 7 e^{- 4 x}) - (- 16 C)}{16 e^{- 4 x}}$

Step 8.2.3.14

Factor $- 1$ out of $- (4 e^{- 4 x} x - 7 e^{- 4 x}) - (- 16 C)$ .

$y = \frac{- (4 e^{- 4 x} x - 7 e^{- 4 x} - 16 C)}{16 e^{- 4 x}}$

Step 8.2.3.15

Simplify the expression.

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Step 8.2.3.15.1

Rewrite $- (4 e^{- 4 x} x - 7 e^{- 4 x} - 16 C)$ as $- 1 (4 e^{- 4 x} x - 7 e^{- 4 x} - 16 C)$ .

$y = \frac{- 1 (4 e^{- 4 x} x - 7 e^{- 4 x} - 16 C)}{16 e^{- 4 x}}$

Step 8.2.3.15.2

Move the negative in front of the fraction.

$y = - \frac{4 e^{- 4 x} x - 7 e^{- 4 x} - 16 C}{16 e^{- 4 x}}$

Step 8.2.3.15.3

Reorder factors in $- \frac{4 e^{- 4 x} x - 7 e^{- 4 x} - 16 C}{16 e^{- 4 x}}$ .

$y = - \frac{4 x e^{- 4 x} - 7 e^{- 4 x} - 16 C}{16 e^{- 4 x}}$