Calculus Examples

$\frac{d y}{d x} + 4 y = 6 \sin (6 x)$

Step 1

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

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Step 1.1

Set up the integration.

$e^{\int 4 d x}$

Step 1.2

Apply the constant rule.

$e^{4 x + C}$

Step 1.3

Remove the constant of integration.

$e^{4 x}$

Step 2

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

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Step 2.1

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

$e^{4 x} \frac{d y}{d x} + e^{4 x} (4 y) = e^{4 x} (6 \sin (6 x))$

Step 2.2

Rewrite using the commutative property of multiplication.

$e^{4 x} \frac{d y}{d x} + 4 e^{4 x} y = e^{4 x} (6 \sin (6 x))$

Step 2.3

Rewrite using the commutative property of multiplication.

$e^{4 x} \frac{d y}{d x} + 4 e^{4 x} y = 6 e^{4 x} \sin (6 x)$

Step 2.4

Reorder factors in $e^{4 x} \frac{d y}{d x} + 4 e^{4 x} y = 6 e^{4 x} \sin (6 x)$ .

$e^{4 x} \frac{d y}{d x} + 4 y e^{4 x} = 6 e^{4 x} \sin (6 x)$

Step 3

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

$\frac{d}{d x} [e^{4 x} y] = 6 e^{4 x} \sin (6 x)$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{4 x} y] d x = \int 6 e^{4 x} \sin (6 x) d x$

Step 5

Integrate the left side.

$e^{4 x} y = \int 6 e^{4 x} \sin (6 x) d x$

Step 6

Integrate the right side.

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Step 6.1

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

$e^{4 x} y = 6 \int e^{4 x} \sin (6 x) d x$

Step 6.2

Reorder $e^{4 x}$ and $\sin (6 x)$ .

$e^{4 x} y = 6 \int \sin (6 x) e^{4 x} d x$

Step 6.3

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \sin (6 x)$ and $d v = e^{4 x}$ .

$e^{4 x} y = 6 (\sin (6 x) (\frac{1}{4} e^{4 x}) - \int \frac{1}{4} e^{4 x} (6 \cos (6 x)) d x)$

Step 6.4

Since $\frac{1}{4} \cdot 6$ is constant with respect to $x$ , move $\frac{1}{4} \cdot 6$ out of the integral.

$e^{4 x} y = 6 (\sin (6 x) (\frac{1}{4} e^{4 x}) - (\frac{1}{4} \cdot 6 \int e^{4 x} (\cos (6 x)) d x))$

Step 6.5

Combine fractions.

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Step 6.5.1

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

$e^{4 x} y = 6 (\sin (6 x) \frac{e^{4 x}}{4} - (\frac{1}{4} \cdot 6 \int e^{4 x} (\cos (6 x)) d x))$

Step 6.5.2

Combine $\sin (6 x)$ and $\frac{e^{4 x}}{4}$ .

$e^{4 x} y = 6 (\frac{\sin (6 x) e^{4 x}}{4} - (\frac{1}{4} \cdot 6 \int e^{4 x} (\cos (6 x)) d x))$

Step 6.5.3

Combine $\frac{1}{4}$ and $6$ .

$e^{4 x} y = 6 (\frac{\sin (6 x) e^{4 x}}{4} - (\frac{6}{4} \int e^{4 x} (\cos (6 x)) d x))$

Step 6.5.4

Reorder $e^{4 x}$ and $\cos (6 x)$ .

$e^{4 x} y = 6 (\frac{\sin (6 x) e^{4 x}}{4} - (\frac{6}{4} \int \cos (6 x) e^{4 x} d x))$

Step 6.6

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \cos (6 x)$ and $d v = e^{4 x}$ .

$e^{4 x} y = 6 (\frac{\sin (6 x) e^{4 x}}{4} - (\frac{6}{4} (\cos (6 x) (\frac{1}{4} e^{4 x}) - \int \frac{1}{4} e^{4 x} (- 6 \sin (6 x)) d x)))$

Step 6.7

Since $\frac{1}{4} \cdot - 6$ is constant with respect to $x$ , move $\frac{1}{4} \cdot - 6$ out of the integral.

$e^{4 x} y = 6 (\frac{\sin (6 x) e^{4 x}}{4} - (\frac{6}{4} (\cos (6 x) (\frac{1}{4} e^{4 x}) - (\frac{1}{4} \cdot - 6 \int e^{4 x} (\sin (6 x)) d x))))$

Step 6.8

Simplify terms.

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Step 6.8.1

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

$e^{4 x} y = 6 (\frac{\sin (6 x) e^{4 x}}{4} - (\frac{6}{4} (\cos (6 x) \frac{e^{4 x}}{4} - (\frac{1}{4} \cdot - 6 \int e^{4 x} (\sin (6 x)) d x))))$

Step 6.8.2

Combine $\cos (6 x)$ and $\frac{e^{4 x}}{4}$ .

$e^{4 x} y = 6 (\frac{\sin (6 x) e^{4 x}}{4} - (\frac{6}{4} (\frac{\cos (6 x) e^{4 x}}{4} - (\frac{1}{4} \cdot - 6 \int e^{4 x} (\sin (6 x)) d x))))$

Step 6.8.3

Combine $\frac{1}{4}$ and $- 6$ .

$e^{4 x} y = 6 (\frac{\sin (6 x) e^{4 x}}{4} - (\frac{6}{4} (\frac{\cos (6 x) e^{4 x}}{4} - (\frac{- 6}{4} \int e^{4 x} (\sin (6 x)) d x))))$

Step 6.8.4

Apply the distributive property.

$e^{4 x} y = 6 (\frac{\sin (6 x) e^{4 x}}{4} - \frac{6}{4} \cdot \frac{\cos (6 x) e^{4 x}}{4} - \frac{6}{4} (- (\frac{- 6}{4} \int e^{4 x} (\sin (6 x)) d x)))$

Step 6.8.5

Multiply $\frac{\cos (6 x) e^{4 x}}{4}$ by $\frac{6}{4}$ .

$e^{4 x} y = 6 (\frac{\sin (6 x) e^{4 x}}{4} - \frac{\cos (6 x) e^{4 x} \cdot 6}{4 \cdot 4} - \frac{6}{4} (- (\frac{- 6}{4} \int e^{4 x} (\sin (6 x)) d x)))$

Step 6.8.6

Multiply.

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Step 6.8.6.1

Multiply $4$ by $4$ .

$e^{4 x} y = 6 (\frac{\sin (6 x) e^{4 x}}{4} - \frac{\cos (6 x) e^{4 x} \cdot 6}{16} - \frac{6}{4} (- (\frac{- 6}{4} \int e^{4 x} (\sin (6 x)) d x)))$

Step 6.8.6.2

Multiply $- 1$ by $- 1$ .

$e^{4 x} y = 6 (\frac{\sin (6 x) e^{4 x}}{4} - \frac{\cos (6 x) e^{4 x} \cdot 6}{16} + 1 (\frac{6}{4}) (\frac{- 6}{4} \int e^{4 x} (\sin (6 x)) d x))$

Step 6.8.6.3

Multiply $\frac{6}{4}$ by $1$ .

$e^{4 x} y = 6 (\frac{\sin (6 x) e^{4 x}}{4} - \frac{\cos (6 x) e^{4 x} \cdot 6}{16} + \frac{6}{4} (\frac{- 6}{4} \int e^{4 x} (\sin (6 x)) d x))$

Step 6.8.7

Multiply $\frac{- 6}{4}$ by $\frac{6}{4}$ .

$e^{4 x} y = 6 (\frac{\sin (6 x) e^{4 x}}{4} - \frac{\cos (6 x) e^{4 x} \cdot 6}{16} + \frac{- 6 \cdot 6}{4 \cdot 4} \int e^{4 x} (\sin (6 x)) d x)$

Step 6.8.8

Multiply.

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Step 6.8.8.1

Multiply $- 6$ by $6$ .

$e^{4 x} y = 6 (\frac{\sin (6 x) e^{4 x}}{4} - \frac{\cos (6 x) e^{4 x} \cdot 6}{16} + \frac{- 36}{4 \cdot 4} \int e^{4 x} (\sin (6 x)) d x)$

Step 6.8.8.2

Multiply $4$ by $4$ .

$e^{4 x} y = 6 (\frac{\sin (6 x) e^{4 x}}{4} - \frac{\cos (6 x) e^{4 x} \cdot 6}{16} + \frac{- 36}{16} \int e^{4 x} (\sin (6 x)) d x)$

Step 6.9

Solving for $\int e^{4 x} \sin (6 x) d x$ , we find that $\int e^{4 x} \sin (6 x) d x$ = $\frac{\frac{\sin (6 x) e^{4 x}}{4} - \frac{\cos (6 x) e^{4 x} \cdot 6}{16}}{\frac{52}{16}}$ .

$e^{4 x} y = 6 (\frac{\frac{\sin (6 x) e^{4 x}}{4} - \frac{\cos (6 x) e^{4 x} \cdot 6}{16}}{\frac{52}{16}} + C)$

Step 6.10

Simplify the answer.

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Step 6.10.1

Simplify.

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Step 6.10.1.1

Move $6$ to the left of $\cos (6 x) e^{4 x}$ .

$e^{4 x} y = 6 (\frac{\frac{\sin (6 x) e^{4 x}}{4} - \frac{6 \cdot (\cos (6 x) e^{4 x})}{16}}{\frac{52}{16}} + C)$

Step 6.10.1.2

Cancel the common factor of $6$ and $16$ .

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Step 6.10.1.2.1

Factor $2$ out of $6 \cos (6 x) e^{4 x}$ .

$e^{4 x} y = 6 (\frac{\frac{\sin (6 x) e^{4 x}}{4} - \frac{2 (3 \cos (6 x) e^{4 x})}{16}}{\frac{52}{16}} + C)$

Step 6.10.1.2.2

Cancel the common factors.

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Step 6.10.1.2.2.1

Factor $2$ out of $16$ .

$e^{4 x} y = 6 (\frac{\frac{\sin (6 x) e^{4 x}}{4} - \frac{2 (3 \cos (6 x) e^{4 x})}{2 (8)}}{\frac{52}{16}} + C)$

Step 6.10.1.2.2.2

Cancel the common factor.

$e^{4 x} y = 6 (\frac{\frac{\sin (6 x) e^{4 x}}{4} - \frac{2 (3 \cos (6 x) e^{4 x})}{2 \cdot 8}}{\frac{52}{16}} + C)$

Step 6.10.1.2.2.3

Rewrite the expression.

$e^{4 x} y = 6 (\frac{\frac{\sin (6 x) e^{4 x}}{4} - \frac{3 \cos (6 x) e^{4 x}}{8}}{\frac{52}{16}} + C)$

Step 6.10.1.3

Cancel the common factor of $52$ and $16$ .

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Step 6.10.1.3.1

Factor $4$ out of $52$ .

$e^{4 x} y = 6 (\frac{\frac{\sin (6 x) e^{4 x}}{4} - \frac{3 \cos (6 x) e^{4 x}}{8}}{\frac{4 (13)}{16}} + C)$

Step 6.10.1.3.2

Cancel the common factors.

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Step 6.10.1.3.2.1

Factor $4$ out of $16$ .

$e^{4 x} y = 6 (\frac{\frac{\sin (6 x) e^{4 x}}{4} - \frac{3 \cos (6 x) e^{4 x}}{8}}{\frac{4 \cdot 13}{4 \cdot 4}} + C)$

Step 6.10.1.3.2.2

Cancel the common factor.

$e^{4 x} y = 6 (\frac{\frac{\sin (6 x) e^{4 x}}{4} - \frac{3 \cos (6 x) e^{4 x}}{8}}{\frac{4 \cdot 13}{4 \cdot 4}} + C)$

Step 6.10.1.3.2.3

Rewrite the expression.

$e^{4 x} y = 6 (\frac{\frac{\sin (6 x) e^{4 x}}{4} - \frac{3 \cos (6 x) e^{4 x}}{8}}{\frac{13}{4}} + C)$

Step 6.10.1.4

Multiply by the reciprocal of the fraction to divide by $\frac{13}{4}$ .

$e^{4 x} y = 6 ((\frac{\sin (6 x) e^{4 x}}{4} - \frac{3 \cos (6 x) e^{4 x}}{8}) \frac{4}{13} + C)$

Step 6.10.2

Rewrite $6 ((\frac{\sin (6 x) e^{4 x}}{4} - \frac{3 \cos (6 x) e^{4 x}}{8}) \frac{4}{13} + C)$ as $6 (\frac{\sin (6 x) e^{4 x} \cdot 4}{4 \cdot 13} + \frac{- 3 \cos (6 x) e^{4 x}}{26}) + C$ .

$e^{4 x} y = 6 (\frac{\sin (6 x) e^{4 x} \cdot 4}{4 \cdot 13} + \frac{- 3 \cos (6 x) e^{4 x}}{26}) + C$

Step 6.10.3

Simplify.

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Step 6.10.3.1

Move $4$ to the left of $\sin (6 x) e^{4 x}$ .

$e^{4 x} y = 6 (\frac{4 \cdot (\sin (6 x) e^{4 x})}{4 \cdot 13} + \frac{- 3 \cos (6 x) e^{4 x}}{26}) + C$

Step 6.10.3.2

Multiply $4$ by $13$ .

$e^{4 x} y = 6 (\frac{4 \sin (6 x) e^{4 x}}{52} + \frac{- 3 \cos (6 x) e^{4 x}}{26}) + C$

Step 6.10.3.3

Cancel the common factor of $4$ and $52$ .

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Step 6.10.3.3.1

Factor $4$ out of $4 \sin (6 x) e^{4 x}$ .

$e^{4 x} y = 6 (\frac{4 (\sin (6 x) e^{4 x})}{52} + \frac{- 3 \cos (6 x) e^{4 x}}{26}) + C$

Step 6.10.3.3.2

Cancel the common factors.

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Step 6.10.3.3.2.1

Factor $4$ out of $52$ .

$e^{4 x} y = 6 (\frac{4 (\sin (6 x) e^{4 x})}{4 \cdot 13} + \frac{- 3 \cos (6 x) e^{4 x}}{26}) + C$

Step 6.10.3.3.2.2

Cancel the common factor.

$e^{4 x} y = 6 (\frac{4 (\sin (6 x) e^{4 x})}{4 \cdot 13} + \frac{- 3 \cos (6 x) e^{4 x}}{26}) + C$

Step 6.10.3.3.2.3

Rewrite the expression.

$e^{4 x} y = 6 (\frac{\sin (6 x) e^{4 x}}{13} + \frac{- 3 \cos (6 x) e^{4 x}}{26}) + C$

Step 6.10.3.4

Move the negative in front of the fraction.

$e^{4 x} y = 6 (\frac{\sin (6 x) e^{4 x}}{13} - \frac{3 \cos (6 x) e^{4 x}}{26}) + C$

Step 6.10.4

Simplify.

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Step 6.10.4.1

Reorder factors in $\frac{\sin (6 x) e^{4 x}}{13}$ .

$e^{4 x} y = 6 (\frac{e^{4 x} \sin (6 x)}{13} - \frac{3 \cos (6 x) e^{4 x}}{26}) + C$

Step 6.10.4.2

Reorder factors in $\frac{3 \cos (6 x) e^{4 x}}{26}$ .

$e^{4 x} y = 6 (\frac{1}{13} e^{4 x} \sin (6 x) - \frac{3}{26} e^{4 x} \cos (6 x)) + C$

Step 7

Solve for $y$ .

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Step 7.1

Simplify.

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Step 7.1.1

Combine $e^{4 x}$ and $\frac{3}{26}$ .

$e^{4 x} y = 6 (\frac{1}{13} \cdot (e^{4 x} \sin (6 x)) - \frac{e^{4 x} \cdot 3}{26} \cdot \cos (6 x)) + C$

Step 7.1.2

Combine $\cos (6 x)$ and $\frac{e^{4 x} \cdot 3}{26}$ .

$e^{4 x} y = 6 (\frac{1}{13} \cdot (e^{4 x} \sin (6 x)) - \frac{\cos (6 x) (e^{4 x} \cdot 3)}{26}) + C$

Step 7.1.3

Remove parentheses.

$e^{4 x} y = 6 (\frac{1}{13} \cdot e^{4 x} \sin (6 x) - \frac{\cos (6 x) (e^{4 x} \cdot 3)}{26}) + C$

Step 7.2

Divide each term in $e^{4 x} y = 6 (\frac{1}{13} \cdot (e^{4 x} \sin (6 x)) - \frac{\cos (6 x) (e^{4 x} \cdot 3)}{26}) + C$ by $e^{4 x}$ and simplify.

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Step 7.2.1

Divide each term in $e^{4 x} y = 6 (\frac{1}{13} \cdot (e^{4 x} \sin (6 x)) - \frac{\cos (6 x) (e^{4 x} \cdot 3)}{26}) + C$ by $e^{4 x}$ .

$\frac{e^{4 x} y}{e^{4 x}} = \frac{6 (\frac{1}{13} \cdot (e^{4 x} \sin (6 x)) - \frac{\cos (6 x) (e^{4 x} \cdot 3)}{26})}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

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

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{e^{4 x} y}{e^{4 x}} = \frac{6 (\frac{1}{13} \cdot (e^{4 x} \sin (6 x)) - \frac{\cos (6 x) (e^{4 x} \cdot 3)}{26})}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{6 (\frac{1}{13} \cdot (e^{4 x} \sin (6 x)) - \frac{\cos (6 x) (e^{4 x} \cdot 3)}{26})}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Simplify each term.

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Step 7.2.3.1.1

Simplify the numerator.

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Step 7.2.3.1.1.1

Factor $e^{4 x}$ out of $\frac{1}{13} \cdot e^{4 x} \sin (6 x) - \frac{\cos (6 x) e^{4 x} \cdot 3}{26}$ .

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Step 7.2.3.1.1.1.1

Factor $e^{4 x}$ out of $\frac{1}{13} \cdot e^{4 x} \sin (6 x)$ .

$y = \frac{6 (e^{4 x} (\frac{1}{13} \sin (6 x)) - \frac{\cos (6 x) e^{4 x} \cdot 3}{26})}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.2.3.1.1.1.2

Factor $e^{4 x}$ out of $- \frac{\cos (6 x) e^{4 x} \cdot 3}{26}$ .

$y = \frac{6 (e^{4 x} (\frac{1}{13} \sin (6 x)) + e^{4 x} (- \frac{\cos (6 x) \cdot 3}{26}))}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.2.3.1.1.1.3

Factor $e^{4 x}$ out of $e^{4 x} (\frac{1}{13} \sin (6 x)) + e^{4 x} (- \frac{\cos (6 x) \cdot 3}{26})$ .

$y = \frac{6 (e^{4 x} (\frac{1}{13} \sin (6 x) - \frac{\cos (6 x) \cdot 3}{26}))}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.2.3.1.1.2

Combine $\frac{1}{13}$ and $\sin (6 x)$ .

$y = \frac{6 (e^{4 x} (\frac{\sin (6 x)}{13} - \frac{\cos (6 x) \cdot 3}{26}))}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.2.3.1.1.3

Move $3$ to the left of $\cos (6 x)$ .

$y = \frac{6 (e^{4 x} (\frac{\sin (6 x)}{13} - \frac{3 \cdot \cos (6 x)}{26}))}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.2.3.1.1.4

Remove unnecessary parentheses.

$y = \frac{6 e^{4 x} (\frac{\sin (6 x)}{13} - \frac{3 \cos (6 x)}{26})}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.2.3.1.2

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

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Step 7.2.3.1.2.1

Cancel the common factor.

$y = \frac{6 e^{4 x} (\frac{\sin (6 x)}{13} - \frac{3 \cos (6 x)}{26})}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.2.3.1.2.2

Divide $6 (\frac{\sin (6 x)}{13} - \frac{3 \cos (6 x)}{26})$ by $1$ .

$y = 6 (\frac{\sin (6 x)}{13} - \frac{3 \cos (6 x)}{26}) + \frac{C}{e^{4 x}}$

Step 7.2.3.1.3

Apply the distributive property.

$y = 6 \frac{\sin (6 x)}{13} + 6 (- \frac{3 \cos (6 x)}{26}) + \frac{C}{e^{4 x}}$

Step 7.2.3.1.4

Combine $6$ and $\frac{\sin (6 x)}{13}$ .

$y = \frac{6 \sin (6 x)}{13} + 6 (- \frac{3 \cos (6 x)}{26}) + \frac{C}{e^{4 x}}$

Step 7.2.3.1.5

Cancel the common factor of $2$ .

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Step 7.2.3.1.5.1

Move the leading negative in $- \frac{3 \cos (6 x)}{26}$ into the numerator.

$y = \frac{6 \sin (6 x)}{13} + 6 \frac{- 3 \cos (6 x)}{26} + \frac{C}{e^{4 x}}$

Step 7.2.3.1.5.2

Factor $2$ out of $6$ .

$y = \frac{6 \sin (6 x)}{13} + 2 (3) \frac{- 3 \cos (6 x)}{26} + \frac{C}{e^{4 x}}$

Step 7.2.3.1.5.3

Factor $2$ out of $26$ .

$y = \frac{6 \sin (6 x)}{13} + 2 \cdot 3 \frac{- 3 \cos (6 x)}{2 \cdot 13} + \frac{C}{e^{4 x}}$

Step 7.2.3.1.5.4

Cancel the common factor.

$y = \frac{6 \sin (6 x)}{13} + 2 \cdot 3 \frac{- 3 \cos (6 x)}{2 \cdot 13} + \frac{C}{e^{4 x}}$

Step 7.2.3.1.5.5

Rewrite the expression.

$y = \frac{6 \sin (6 x)}{13} + 3 \frac{- 3 \cos (6 x)}{13} + \frac{C}{e^{4 x}}$

Step 7.2.3.1.6

Combine $3$ and $\frac{- 3 \cos (6 x)}{13}$ .

$y = \frac{6 \sin (6 x)}{13} + \frac{3 (- 3 \cos (6 x))}{13} + \frac{C}{e^{4 x}}$

Step 7.2.3.1.7

Multiply $- 3$ by $3$ .

$y = \frac{6 \sin (6 x)}{13} + \frac{- 9 \cos (6 x)}{13} + \frac{C}{e^{4 x}}$

Step 7.2.3.1.8

Move the negative in front of the fraction.

$y = \frac{6 \sin (6 x)}{13} - \frac{9 \cos (6 x)}{13} + \frac{C}{e^{4 x}}$