Calculus Examples

$\frac{d y}{d x} = 3 (y + 2 x) + 1$

Step 1

Rewrite the equation as $M (x) \frac{d y}{d x} + P (x) y = Q (x)$ .

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Step 1.1

Apply the distributive property.

$\frac{d y}{d x} = 3 y + 3 (2 x) + 1$

Step 1.2

Subtract $3 y$ from both sides of the equation.

$\frac{d y}{d x} - 3 y = 3 (2 x) + 1$

Step 1.3

Reorder terms.

$\frac{d y}{d x} - 3 y = 2 \cdot 3 x + 1$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = - 3$ .

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Step 2.1

Set up the integration.

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

Step 2.2

Apply the constant rule.

$e^{- 3 x + C}$

Step 2.3

Remove the constant of integration.

$e^{- 3 x}$

Step 3

Multiply each term by the integrating factor $e^{- 3 x}$ .

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Step 3.1

Multiply each term by $e^{- 3 x}$ .

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Simplify each term.

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Step 3.3.1

Rewrite using the commutative property of multiplication.

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

Step 3.3.2

Multiply $2$ by $3$ .

$e^{- 3 x} \frac{d y}{d x} - 3 e^{- 3 x} y = 6 e^{- 3 x} x + e^{- 3 x} \cdot 1$

Step 3.3.3

Multiply $e^{- 3 x}$ by $1$ .

$e^{- 3 x} \frac{d y}{d x} - 3 e^{- 3 x} y = 6 e^{- 3 x} x + e^{- 3 x}$

Step 3.4

Reorder factors in $e^{- 3 x} \frac{d y}{d x} - 3 e^{- 3 x} y = 6 e^{- 3 x} x + e^{- 3 x}$ .

$e^{- 3 x} \frac{d y}{d x} - 3 y e^{- 3 x} = 6 x e^{- 3 x} + e^{- 3 x}$

Step 4

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

$\frac{d}{d x} [e^{- 3 x} y] = 6 x e^{- 3 x} + e^{- 3 x}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [e^{- 3 x} y] d x = \int 6 x e^{- 3 x} + e^{- 3 x} d x$

Step 6

Integrate the left side.

$e^{- 3 x} y = \int 6 x e^{- 3 x} + e^{- 3 x} d x$

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

$e^{- 3 x} y = \int 6 x e^{- 3 x} d x + \int e^{- 3 x} d x$

Step 7.2

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$e^{- 3 x} y = 6 \int x e^{- 3 x} d x + \int e^{- 3 x} d x$

Step 7.3

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{- 3 x}$ .

$e^{- 3 x} y = 6 (x (- \frac{1}{3} e^{- 3 x}) - \int - \frac{1}{3} e^{- 3 x} d x) + \int e^{- 3 x} d x$

Step 7.4

Simplify.

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Step 7.4.1

Combine $e^{- 3 x}$ and $\frac{1}{3}$ .

$e^{- 3 x} y = 6 (x (- \frac{e^{- 3 x}}{3}) - \int - \frac{1}{3} e^{- 3 x} d x) + \int e^{- 3 x} d x$

Step 7.4.2

Combine $x$ and $\frac{e^{- 3 x}}{3}$ .

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} - \int - \frac{1}{3} e^{- 3 x} d x) + \int e^{- 3 x} d x$

Step 7.4.3

Combine $e^{- 3 x}$ and $\frac{1}{3}$ .

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} - \int - \frac{e^{- 3 x}}{3} d x) + \int e^{- 3 x} d x$

Step 7.5

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

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} - - \int \frac{e^{- 3 x}}{3} d x) + \int e^{- 3 x} d x$

Step 7.6

Simplify.

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Step 7.6.1

Multiply $- 1$ by $- 1$ .

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} + 1 \int \frac{e^{- 3 x}}{3} d x) + \int e^{- 3 x} d x$

Step 7.6.2

Multiply $\int \frac{e^{- 3 x}}{3} d x$ by $1$ .

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} + \int \frac{e^{- 3 x}}{3} d x) + \int e^{- 3 x} d x$

Step 7.7

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} + \frac{1}{3} \int e^{- 3 x} d x) + \int e^{- 3 x} d x$

Step 7.8

Let $u_{1} = - 3 x$ . Then $d u_{1} = - 3 d x$ , so $- \frac{1}{3} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 7.8.1

Let $u_{1} = - 3 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 7.8.1.1

Differentiate $- 3 x$ .

$\frac{d}{d x} [- 3 x]$

Step 7.8.1.2

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$- 3 \frac{d}{d x} [x]$

Step 7.8.1.3

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

$- 3 \cdot 1$

Step 7.8.1.4

Multiply $- 3$ by $1$ .

$- 3$

Step 7.8.2

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

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} + \frac{1}{3} \int e^{u_{1}} \frac{1}{- 3} d u_{1}) + \int e^{- 3 x} d x$

Step 7.9

Simplify.

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Step 7.9.1

Move the negative in front of the fraction.

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} + \frac{1}{3} \int e^{u_{1}} (- \frac{1}{3}) d u_{1}) + \int e^{- 3 x} d x$

Step 7.9.2

Combine $e^{u_{1}}$ and $\frac{1}{3}$ .

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} + \frac{1}{3} \int - \frac{e^{u_{1}}}{3} d u_{1}) + \int e^{- 3 x} d x$

Step 7.10

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

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} + \frac{1}{3} (- \int \frac{e^{u_{1}}}{3} d u_{1})) + \int e^{- 3 x} d x$

Step 7.11

Since $\frac{1}{3}$ is constant with respect to $u_{1}$ , move $\frac{1}{3}$ out of the integral.

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} + \frac{1}{3} (- (\frac{1}{3} \int e^{u_{1}} d u_{1}))) + \int e^{- 3 x} d x$

Step 7.12

Simplify.

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Step 7.12.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} - \frac{1}{3 \cdot 3} \int e^{u_{1}} d u_{1}) + \int e^{- 3 x} d x$

Step 7.12.2

Multiply $3$ by $3$ .

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} - \frac{1}{9} \int e^{u_{1}} d u_{1}) + \int e^{- 3 x} d x$

Step 7.13

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

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C)) + \int e^{- 3 x} d x$

Step 7.14

Let $u_{2} = - 3 x$ . Then $d u_{2} = - 3 d x$ , so $- \frac{1}{3} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 7.14.1

Let $u_{2} = - 3 x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 7.14.1.1

Differentiate $- 3 x$ .

$\frac{d}{d x} [- 3 x]$

Step 7.14.1.2

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$- 3 \frac{d}{d x} [x]$

Step 7.14.1.3

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

$- 3 \cdot 1$

Step 7.14.1.4

Multiply $- 3$ by $1$ .

$- 3$

Step 7.14.2

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

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C)) + \int e^{u_{2}} \frac{1}{- 3} d u_{2}$

Step 7.15

Simplify.

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Step 7.15.1

Move the negative in front of the fraction.

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C)) + \int e^{u_{2}} (- \frac{1}{3}) d u_{2}$

Step 7.15.2

Combine $e^{u_{2}}$ and $\frac{1}{3}$ .

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C)) + \int - \frac{e^{u_{2}}}{3} d u_{2}$

Step 7.16

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

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C)) - \int \frac{e^{u_{2}}}{3} d u_{2}$

Step 7.17

Since $\frac{1}{3}$ is constant with respect to $u_{2}$ , move $\frac{1}{3}$ out of the integral.

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C)) - (\frac{1}{3} \int e^{u_{2}} d u_{2})$

Step 7.18

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

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C)) - \frac{1}{3} (e^{u_{2}} + C)$

Step 7.19

Simplify.

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Step 7.19.1

Simplify.

$e^{- 3 x} y = 6 (- \frac{1}{3} x e^{- 3 x} - \frac{1}{9} e^{u_{1}}) - \frac{1}{3} e^{u_{2}} + C$

Step 7.19.2

Simplify.

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Step 7.19.2.1

Combine $x$ and $\frac{1}{3}$ .

$e^{- 3 x} y = 6 (- \frac{x}{3} e^{- 3 x} - \frac{1}{9} e^{u_{1}}) - \frac{1}{3} e^{u_{2}} + C$

Step 7.19.2.2

Combine $e^{- 3 x}$ and $\frac{x}{3}$ .

$e^{- 3 x} y = 6 (- \frac{e^{- 3 x} x}{3} - \frac{1}{9} e^{u_{1}}) - \frac{1}{3} e^{u_{2}} + C$

Step 7.19.2.3

Combine $e^{u_{1}}$ and $\frac{1}{9}$ .

$e^{- 3 x} y = 6 (- \frac{e^{- 3 x} x}{3} - \frac{e^{u_{1}}}{9}) - \frac{1}{3} e^{u_{2}} + C$

Step 7.20

Substitute back in for each integration substitution variable.

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Step 7.20.1

Replace all occurrences of $u_{1}$ with $- 3 x$ .

$e^{- 3 x} y = 6 (- \frac{e^{- 3 x} x}{3} - \frac{e^{- 3 x}}{9}) - \frac{1}{3} e^{u_{2}} + C$

Step 7.20.2

Replace all occurrences of $u_{2}$ with $- 3 x$ .

$e^{- 3 x} y = 6 (- \frac{e^{- 3 x} x}{3} - \frac{e^{- 3 x}}{9}) - \frac{1}{3} e^{- 3 x} + C$

Step 7.21

Combine $e^{- 3 x}$ and $\frac{1}{3}$ .

$e^{- 3 x} y = 6 (- \frac{e^{- 3 x} x}{3} - \frac{e^{- 3 x}}{9}) - \frac{e^{- 3 x}}{3} + C$

Step 7.22

Reorder terms.

$e^{- 3 x} y = 6 (- \frac{1}{3} e^{- 3 x} x - \frac{1}{9} e^{- 3 x}) - \frac{1}{3} e^{- 3 x} + C$

Step 8

Solve for $y$ .

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Step 8.1

Simplify.

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Step 8.1.1

Simplify each term.

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Step 8.1.1.1

Multiply $- \frac{1}{3} (e^{- 3 x} x)$ .

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Step 8.1.1.1.1

Combine $e^{- 3 x}$ and $\frac{1}{3}$ .

$e^{- 3 x} y = 6 (- \frac{e^{- 3 x}}{3} x - \frac{1}{9} \cdot e^{- 3 x}) - \frac{1}{3} \cdot e^{- 3 x} + C$

Step 8.1.1.1.2

Combine $x$ and $\frac{e^{- 3 x}}{3}$ .

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} - \frac{1}{9} \cdot e^{- 3 x}) - \frac{1}{3} \cdot e^{- 3 x} + C$

Step 8.1.1.2

Combine $e^{- 3 x}$ and $\frac{1}{9}$ .

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3} - \frac{e^{- 3 x}}{9}) - \frac{1}{3} \cdot e^{- 3 x} + C$

Step 8.1.2

Apply the distributive property.

$e^{- 3 x} y = 6 (- \frac{x e^{- 3 x}}{3}) + 6 (- \frac{e^{- 3 x}}{9}) - \frac{1}{3} \cdot e^{- 3 x} + C$

Step 8.1.3

Cancel the common factor of $3$ .

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Step 8.1.3.1

Move the leading negative in $- \frac{x e^{- 3 x}}{3}$ into the numerator.

$e^{- 3 x} y = 6 \frac{- x e^{- 3 x}}{3} + 6 (- \frac{e^{- 3 x}}{9}) - \frac{1}{3} \cdot e^{- 3 x} + C$

Step 8.1.3.2

Factor $3$ out of $6$ .

$e^{- 3 x} y = 3 (2) \frac{- x e^{- 3 x}}{3} + 6 (- \frac{e^{- 3 x}}{9}) - \frac{1}{3} \cdot e^{- 3 x} + C$

Step 8.1.3.3

Cancel the common factor.

$e^{- 3 x} y = 3 \cdot 2 \frac{- x e^{- 3 x}}{3} + 6 (- \frac{e^{- 3 x}}{9}) - \frac{1}{3} \cdot e^{- 3 x} + C$

Step 8.1.3.4

Rewrite the expression.

$e^{- 3 x} y = 2 (- x e^{- 3 x}) + 6 (- \frac{e^{- 3 x}}{9}) - \frac{1}{3} \cdot e^{- 3 x} + C$

Step 8.1.4

Multiply $- 1$ by $2$ .

$e^{- 3 x} y = - 2 (x e^{- 3 x}) + 6 (- \frac{e^{- 3 x}}{9}) - \frac{1}{3} \cdot e^{- 3 x} + C$

Step 8.1.5

Cancel the common factor of $3$ .

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Step 8.1.5.1

Move the leading negative in $- \frac{e^{- 3 x}}{9}$ into the numerator.

$e^{- 3 x} y = - 2 (x e^{- 3 x}) + 6 \frac{- e^{- 3 x}}{9} - \frac{1}{3} \cdot e^{- 3 x} + C$

Step 8.1.5.2

Factor $3$ out of $6$ .

$e^{- 3 x} y = - 2 (x e^{- 3 x}) + 3 (2) \frac{- e^{- 3 x}}{9} - \frac{1}{3} \cdot e^{- 3 x} + C$

Step 8.1.5.3

Factor $3$ out of $9$ .

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

Step 8.1.5.4

Cancel the common factor.

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

Step 8.1.5.5

Rewrite the expression.

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

Step 8.1.6

Combine $2$ and $\frac{- e^{- 3 x}}{3}$ .

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

Step 8.1.7

Multiply $- 1$ by $2$ .

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

Step 8.1.8

Move the negative in front of the fraction.

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

Step 8.1.9

To write $- 2 x e^{- 3 x}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 8.1.10

Combine $- 2 x e^{- 3 x}$ and $\frac{3}{3}$ .

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

Step 8.1.11

Combine the numerators over the common denominator.

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

Step 8.1.12

Simplify the numerator.

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Step 8.1.12.1

Factor $2 e^{- 3 x}$ out of $- 2 x e^{- 3 x} \cdot 3 - 2 e^{- 3 x}$ .

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Step 8.1.12.1.1

Factor $2 e^{- 3 x}$ out of $- 2 x e^{- 3 x} \cdot 3$ .

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

Step 8.1.12.1.2

Factor $2 e^{- 3 x}$ out of $- 2 e^{- 3 x}$ .

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

Step 8.1.12.1.3

Factor $2 e^{- 3 x}$ out of $2 e^{- 3 x} (- x \cdot 3) + 2 e^{- 3 x} (- 1)$ .

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

Step 8.1.12.2

Multiply $3$ by $- 1$ .

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

Step 8.1.13

Factor $- 1$ out of $- 3 x$ .

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

Step 8.1.14

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

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

Step 8.1.15

Factor $- 1$ out of $- (3 x) - 1 (1)$ .

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

Step 8.1.16

Rewrite $- (3 x + 1)$ as $- 1 (3 x + 1)$ .

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

Step 8.1.17

Move the negative in front of the fraction.

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

Step 8.1.18

Reorder factors in $- \frac{(2 e^{- 3 x}) (3 x + 1)}{3}$ .

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

Step 8.1.19

Combine $e^{- 3 x}$ and $\frac{1}{3}$ .

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

Step 8.2

Divide each term in $e^{- 3 x} y = - \frac{2 e^{- 3 x} (3 x + 1)}{3} - \frac{e^{- 3 x}}{3} + C$ by $e^{- 3 x}$ and simplify.

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Step 8.2.1

Divide each term in $e^{- 3 x} y = - \frac{2 e^{- 3 x} (3 x + 1)}{3} - \frac{e^{- 3 x}}{3} + C$ by $e^{- 3 x}$ .

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

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

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

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Step 8.2.2.1.1

Cancel the common factor.

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

Step 8.2.2.1.2

Divide $y$ by $1$ .

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

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Combine fractions.

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Step 8.2.3.1.1

Combine the numerators over the common denominator.

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

Step 8.2.3.1.2

Combine the numerators over the common denominator.

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

Step 8.2.3.2

Simplify each term.

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Step 8.2.3.2.1

Apply the distributive property.

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

Step 8.2.3.2.2

Rewrite using the commutative property of multiplication.

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

Step 8.2.3.2.3

Multiply $- 2$ by $1$ .

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

Step 8.2.3.2.4

Multiply $- 2$ by $3$ .

$y = \frac{C + \frac{- 6 e^{- 3 x} x - 2 e^{- 3 x} - e^{- 3 x}}{3}}{e^{- 3 x}}$

Step 8.2.3.3

Simplify by adding terms.

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Step 8.2.3.3.1

Subtract $e^{- 3 x}$ from $- 2 e^{- 3 x}$ .

$y = \frac{C + \frac{- 6 e^{- 3 x} x - 3 e^{- 3 x}}{3}}{e^{- 3 x}}$

Step 8.2.3.3.2

Reorder factors in $- 6 e^{- 3 x} x - 3 e^{- 3 x}$ .

$y = \frac{C + \frac{- 6 x e^{- 3 x} - 3 e^{- 3 x}}{3}}{e^{- 3 x}}$

Step 8.2.3.4

Simplify each term.

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Step 8.2.3.4.1

Factor $3 e^{- 3 x}$ out of $- 6 x e^{- 3 x} - 3 e^{- 3 x}$ .

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Step 8.2.3.4.1.1

Factor $3 e^{- 3 x}$ out of $- 6 x e^{- 3 x}$ .

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

Step 8.2.3.4.1.2

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

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

Step 8.2.3.4.1.3

Factor $3 e^{- 3 x}$ out of $3 e^{- 3 x} (- 2 x) + 3 e^{- 3 x} (- 1)$ .

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

Step 8.2.3.4.2

Cancel the common factor of $3$ .

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Step 8.2.3.4.2.1

Cancel the common factor.

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

Step 8.2.3.4.2.2

Divide $e^{- 3 x} (- 2 x - 1)$ by $1$ .

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

Step 8.2.3.4.3

Apply the distributive property.

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

Step 8.2.3.4.4

Rewrite using the commutative property of multiplication.

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

Step 8.2.3.4.5

Move $- 1$ to the left of $e^{- 3 x}$ .

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

Step 8.2.3.4.6

Rewrite $- 1 e^{- 3 x}$ as $- e^{- 3 x}$ .

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

Step 8.2.3.5

Reorder factors in $C - 2 e^{- 3 x} x - e^{- 3 x}$ .

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