Calculus Examples

$\frac{d y}{d x} = 2 x - 3 y$ , $y (0) = \frac{1}{3}$

Step 1

Add $3 y$ to both sides of the equation.

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

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 $e^{3 x}$ .

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

Reorder factors in $e^{3 x} \frac{d y}{d x} + 3 e^{3 x} y = 2 e^{3 x} x$ .

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

Step 4

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

$\frac{d}{d x} [e^{3 x} y] = 2 x e^{3 x}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [e^{3 x} y] d x = \int 2 x e^{3 x} d x$

Step 6

Integrate the left side.

$e^{3 x} y = \int 2 x e^{3 x} d x$

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.

$e^{3 x} y = 2 \int x e^{3 x} d x$

Step 7.2

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

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

Step 7.3

Simplify.

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Step 7.3.1

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

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

Step 7.3.2

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

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

Step 7.3.3

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

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

Step 7.4

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

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

Step 7.5

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

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Step 7.5.1

Let $u = 3 x$ . Find $\frac{d u}{d x}$ .

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Step 7.5.1.1

Differentiate $3 x$ .

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

Step 7.5.1.2

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 7.5.1.3

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 7.5.1.4

Multiply $3$ by $1$ .

$3$

Step 7.5.2

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

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

Step 7.6

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

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

Step 7.7

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

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

Step 7.8

Simplify.

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Step 7.8.1

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

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

Step 7.8.2

Multiply $3$ by $3$ .

$e^{3 x} y = 2 (\frac{x e^{3 x}}{3} - \frac{1}{9} \int e^{u} d u)$

Step 7.9

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

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

Step 7.10

Rewrite $2 (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u} + C))$ as $2 (\frac{1}{3} x e^{3 x} - \frac{1}{9} e^{u}) + C$ .

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

Step 7.11

Replace all occurrences of $u$ with $3 x$ .

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

Step 8

Solve for $y$ .

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Step 8.1

Simplify.

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Step 8.1.1

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

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

Step 8.1.2

Remove parentheses.

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

Step 8.2

Divide each term in $e^{3 x} y = 2 (\frac{1}{3} \cdot (x e^{3 x}) - \frac{e^{3 x}}{9}) + C$ by $e^{3 x}$ and simplify.

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Step 8.2.1

Divide each term in $e^{3 x} y = 2 (\frac{1}{3} \cdot (x e^{3 x}) - \frac{e^{3 x}}{9}) + C$ by $e^{3 x}$ .

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

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.

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

Step 8.2.2.1.2

Divide $y$ by $1$ .

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

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Simplify each term.

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Step 8.2.3.1.1

Simplify the numerator.

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Step 8.2.3.1.1.1

Factor $e^{3 x}$ out of $\frac{1}{3} \cdot x e^{3 x} - \frac{e^{3 x}}{9}$ .

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Step 8.2.3.1.1.1.1

Factor $e^{3 x}$ out of $\frac{1}{3} \cdot x e^{3 x}$ .

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

Step 8.2.3.1.1.1.2

Factor $e^{3 x}$ out of $- \frac{e^{3 x}}{9}$ .

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

Step 8.2.3.1.1.1.3

Factor $e^{3 x}$ out of $e^{3 x} (\frac{1}{3} \cdot x) + e^{3 x} (- \frac{1}{9})$ .

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

Step 8.2.3.1.1.2

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

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

Step 8.2.3.1.1.3

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

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

Step 8.2.3.1.1.4

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.2.3.1.1.4.1

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

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

Step 8.2.3.1.1.4.2

Multiply $3$ by $3$ .

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

Step 8.2.3.1.1.5

Combine the numerators over the common denominator.

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

Step 8.2.3.1.1.6

Move $3$ to the left of $x$ .

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

Step 8.2.3.1.1.7

Combine exponents.

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Step 8.2.3.1.1.7.1

Combine $\frac{3 x - 1}{9}$ and $2$ .

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

Step 8.2.3.1.1.7.2

Combine $\frac{(3 x - 1) \cdot 2}{9}$ and $e^{3 x}$ .

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

Step 8.2.3.1.1.8

Move $2$ to the left of $3 x - 1$ .

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

Step 8.2.3.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2.3.1.3

Combine.

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

Step 8.2.3.1.4

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

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Step 8.2.3.1.4.1

Cancel the common factor.

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

Step 8.2.3.1.4.2

Rewrite the expression.

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

Step 8.2.3.1.5

Multiply $2$ by $1$ .

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

Step 8.2.3.2

To write $\frac{2 (3 x - 1)}{9}$ as a fraction with a common denominator, multiply by $\frac{e^{3 x}}{e^{3 x}}$ .

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

Step 8.2.3.3

To write $\frac{C}{e^{3 x}}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

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

Step 8.2.3.4

Write each expression with a common denominator of $9 e^{3 x}$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.2.3.4.1

Multiply $\frac{2 (3 x - 1)}{9}$ by $\frac{e^{3 x}}{e^{3 x}}$ .

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

Step 8.2.3.4.2

Multiply $\frac{C}{e^{3 x}}$ by $\frac{9}{9}$ .

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

Step 8.2.3.4.3

Reorder the factors of $e^{3 x} \cdot 9$ .

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

Step 8.2.3.5

Combine the numerators over the common denominator.

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

Step 8.2.3.6

Simplify the numerator.

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Step 8.2.3.6.1

Apply the distributive property.

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

Step 8.2.3.6.2

Multiply $3$ by $2$ .

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

Step 8.2.3.6.3

Multiply $2$ by $- 1$ .

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

Step 8.2.3.6.4

Apply the distributive property.

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

Step 8.2.3.6.5

Move $9$ to the left of $C$ .

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

Step 9

Use the initial condition to find the value of $C$ by substituting $0$ for $x$ and $\frac{1}{3}$ for $y$ in $y = \frac{6 x e^{3 x} - 2 e^{3 x} + 9 C}{9 e^{3 x}}$ .

$\frac{1}{3} = \frac{6 \cdot 0 e^{3 \cdot 0} - 2 e^{3 \cdot 0} + 9 C}{9 e^{3 \cdot 0}}$

Step 10

Solve for $C$ .

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Step 10.1

Rewrite the equation as $\frac{6 \cdot (0 e^{3 \cdot 0}) - 2 e^{3 \cdot 0} + 9 C}{9 e^{3 \cdot 0}} = \frac{1}{3}$ .

$\frac{6 \cdot (0 e^{3 \cdot 0}) - 2 e^{3 \cdot 0} + 9 C}{9 e^{3 \cdot 0}} = \frac{1}{3}$

Step 10.2

Multiply both sides by $9 e^{3 \cdot 0}$ .

$\frac{6 \cdot (0 e^{3 \cdot 0}) - 2 e^{3 \cdot 0} + 9 C}{9 e^{3 \cdot 0}} (9 e^{3 \cdot 0}) = \frac{1}{3} (9 e^{3 \cdot 0})$

Step 10.3

Simplify.

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Step 10.3.1

Simplify the left side.

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Step 10.3.1.1

Simplify $\frac{6 \cdot (0 e^{3 \cdot 0}) - 2 e^{3 \cdot 0} + 9 C}{9 e^{3 \cdot 0}} (9 e^{3 \cdot 0})$ .

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Step 10.3.1.1.1

Cancel the common factor of $9 e^{3 \cdot 0}$ .

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Step 10.3.1.1.1.1

Cancel the common factor.

$\frac{6 \cdot (0 e^{3 \cdot 0}) - 2 e^{3 \cdot 0} + 9 C}{9 e^{3 \cdot 0}} (9 e^{3 \cdot 0}) = \frac{1}{3} (9 e^{3 \cdot 0})$

Step 10.3.1.1.1.2

Rewrite the expression.

$6 \cdot (0 e^{3 \cdot 0}) - 2 e^{3 \cdot 0} + 9 C = \frac{1}{3} (9 e^{3 \cdot 0})$

Step 10.3.1.1.2

Simplify each term.

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Step 10.3.1.1.2.1

Multiply $3$ by $0$ .

$6 \cdot (0 e^{0}) - 2 e^{3 \cdot 0} + 9 C = \frac{1}{3} (9 e^{3 \cdot 0})$

Step 10.3.1.1.2.2

Anything raised to $0$ is $1$ .

$6 \cdot (0 \cdot 1) - 2 e^{3 \cdot 0} + 9 C = \frac{1}{3} (9 e^{3 \cdot 0})$

Step 10.3.1.1.2.3

Multiply $6 (0 \cdot 1)$ .

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Step 10.3.1.1.2.3.1

Multiply $0$ by $1$ .

$6 \cdot 0 - 2 e^{3 \cdot 0} + 9 C = \frac{1}{3} (9 e^{3 \cdot 0})$

Step 10.3.1.1.2.3.2

Multiply $6$ by $0$ .

$0 - 2 e^{3 \cdot 0} + 9 C = \frac{1}{3} (9 e^{3 \cdot 0})$

Step 10.3.1.1.2.4

Multiply $3$ by $0$ .

$0 - 2 e^{0} + 9 C = \frac{1}{3} (9 e^{3 \cdot 0})$

Step 10.3.1.1.2.5

Anything raised to $0$ is $1$ .

$0 - 2 \cdot 1 + 9 C = \frac{1}{3} (9 e^{3 \cdot 0})$

Step 10.3.1.1.2.6

Multiply $- 2$ by $1$ .

$0 - 2 + 9 C = \frac{1}{3} (9 e^{3 \cdot 0})$

Step 10.3.1.1.3

Simplify the expression.

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Step 10.3.1.1.3.1

Subtract $2$ from $0$ .

$- 2 + 9 C = \frac{1}{3} (9 e^{3 \cdot 0})$

Step 10.3.1.1.3.2

Reorder $- 2$ and $9 C$ .

$9 C - 2 = \frac{1}{3} (9 e^{3 \cdot 0})$

Step 10.3.2

Simplify the right side.

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Step 10.3.2.1

Simplify $\frac{1}{3} (9 e^{3 \cdot 0})$ .

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Step 10.3.2.1.1

Cancel the common factor of $3$ .

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Step 10.3.2.1.1.1

Factor $3$ out of $9 e^{3 \cdot 0}$ .

$9 C - 2 = \frac{1}{3} (3 (3 e^{3 \cdot 0}))$

Step 10.3.2.1.1.2

Cancel the common factor.

$9 C - 2 = \frac{1}{3} (3 (3 e^{3 \cdot 0}))$

Step 10.3.2.1.1.3

Rewrite the expression.

$9 C - 2 = 3 e^{3 \cdot 0}$

Step 10.3.2.1.2

Simplify the expression.

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Step 10.3.2.1.2.1

Multiply $3$ by $0$ .

$9 C - 2 = 3 e^{0}$

Step 10.3.2.1.2.2

Anything raised to $0$ is $1$ .

$9 C - 2 = 3 \cdot 1$

Step 10.3.2.1.2.3

Multiply $3$ by $1$ .

$9 C - 2 = 3$

Step 10.4

Solve for $C$ .

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Step 10.4.1

Move all terms not containing $C$ to the right side of the equation.

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Step 10.4.1.1

Add $2$ to both sides of the equation.

$9 C = 3 + 2$

Step 10.4.1.2

Add $3$ and $2$ .

$9 C = 5$

Step 10.4.2

Divide each term in $9 C = 5$ by $9$ and simplify.

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Step 10.4.2.1

Divide each term in $9 C = 5$ by $9$ .

$\frac{9 C}{9} = \frac{5}{9}$

Step 10.4.2.2

Simplify the left side.

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Step 10.4.2.2.1

Cancel the common factor of $9$ .

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Step 10.4.2.2.1.1

Cancel the common factor.

$\frac{9 C}{9} = \frac{5}{9}$

Step 10.4.2.2.1.2

Divide $C$ by $1$ .

$C = \frac{5}{9}$

Step 11

Substitute $\frac{5}{9}$ for $C$ in $y = \frac{6 x e^{3 x} - 2 e^{3 x} + 9 C}{9 e^{3 x}}$ and simplify.

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Step 11.1

Substitute $\frac{5}{9}$ for $C$ .

$y = \frac{6 x e^{3 x} - 2 e^{3 x} + 9 (\frac{5}{9})}{9 e^{3 x}}$

Step 11.2

Simplify the numerator.

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Step 11.2.1

Cancel the common factor of $9$ .

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Step 11.2.1.1

Cancel the common factor.

$y = \frac{6 x e^{3 x} - 2 e^{3 x} + 9 (\frac{5}{9})}{9 e^{3 x}}$

Step 11.2.1.2

Rewrite the expression.

$y = \frac{6 x e^{3 x} - 2 e^{3 x} + 5}{9 e^{3 x}}$

Step 11.2.2

Reorder terms.

$y = \frac{6 e^{3 x} x + 5 - 2 e^{3 x}}{9 e^{3 x}}$

Step 11.3

Reorder factors in $y = \frac{6 e^{3 x} x + 5 - 2 e^{3 x}}{9 e^{3 x}}$ .

$y = \frac{6 x e^{3 x} + 5 - 2 e^{3 x}}{9 e^{3 x}}$