Calculus Examples

$\frac{d y}{d t} + 3 y = 13 \sin (2 t)$ , $y (0) = 6$

Step 1

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

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Step 1.1

Set up the integration.

$e^{\int 3 d t}$

Step 1.2

Apply the constant rule.

$e^{3 t + C}$

Step 1.3

Remove the constant of integration.

$e^{3 t}$

Step 2

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

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Step 2.1

Multiply each term by $e^{3 t}$ .

$e^{3 t} \frac{d y}{d t} + e^{3 t} (3 y) = e^{3 t} (13 \sin (2 t))$

Step 2.2

Rewrite using the commutative property of multiplication.

$e^{3 t} \frac{d y}{d t} + 3 e^{3 t} y = e^{3 t} (13 \sin (2 t))$

Step 2.3

Rewrite using the commutative property of multiplication.

$e^{3 t} \frac{d y}{d t} + 3 e^{3 t} y = 13 e^{3 t} \sin (2 t)$

Step 2.4

Reorder factors in $e^{3 t} \frac{d y}{d t} + 3 e^{3 t} y = 13 e^{3 t} \sin (2 t)$ .

$e^{3 t} \frac{d y}{d t} + 3 y e^{3 t} = 13 e^{3 t} \sin (2 t)$

Step 3

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

$\frac{d}{d t} [e^{3 t} y] = 13 e^{3 t} \sin (2 t)$

Step 4

Set up an integral on each side.

$\int \frac{d}{d t} [e^{3 t} y] d t = \int 13 e^{3 t} \sin (2 t) d t$

Step 5

Integrate the left side.

$e^{3 t} y = \int 13 e^{3 t} \sin (2 t) d t$

Step 6

Integrate the right side.

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Step 6.1

Since $13$ is constant with respect to $t$ , move $13$ out of the integral.

$e^{3 t} y = 13 \int e^{3 t} \sin (2 t) d t$

Step 6.2

Reorder $e^{3 t}$ and $\sin (2 t)$ .

$e^{3 t} y = 13 \int \sin (2 t) e^{3 t} d t$

Step 6.3

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \sin (2 t)$ and $d v = e^{3 t}$ .

$e^{3 t} y = 13 (\sin (2 t) (\frac{1}{3} e^{3 t}) - \int \frac{1}{3} e^{3 t} (2 \cos (2 t)) d t)$

Step 6.4

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

$e^{3 t} y = 13 (\sin (2 t) (\frac{1}{3} e^{3 t}) - (\frac{1}{3} \cdot 2 \int e^{3 t} (\cos (2 t)) d t))$

Step 6.5

Combine fractions.

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Step 6.5.1

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

$e^{3 t} y = 13 (\sin (2 t) \frac{e^{3 t}}{3} - (\frac{1}{3} \cdot 2 \int e^{3 t} (\cos (2 t)) d t))$

Step 6.5.2

Combine $\sin (2 t)$ and $\frac{e^{3 t}}{3}$ .

$e^{3 t} y = 13 (\frac{\sin (2 t) e^{3 t}}{3} - (\frac{1}{3} \cdot 2 \int e^{3 t} (\cos (2 t)) d t))$

Step 6.5.3

Combine $\frac{1}{3}$ and $2$ .

$e^{3 t} y = 13 (\frac{\sin (2 t) e^{3 t}}{3} - (\frac{2}{3} \int e^{3 t} (\cos (2 t)) d t))$

Step 6.5.4

Reorder $e^{3 t}$ and $\cos (2 t)$ .

$e^{3 t} y = 13 (\frac{\sin (2 t) e^{3 t}}{3} - (\frac{2}{3} \int \cos (2 t) e^{3 t} d t))$

Step 6.6

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \cos (2 t)$ and $d v = e^{3 t}$ .

$e^{3 t} y = 13 (\frac{\sin (2 t) e^{3 t}}{3} - (\frac{2}{3} (\cos (2 t) (\frac{1}{3} e^{3 t}) - \int \frac{1}{3} e^{3 t} (- 2 \sin (2 t)) d t)))$

Step 6.7

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

$e^{3 t} y = 13 (\frac{\sin (2 t) e^{3 t}}{3} - (\frac{2}{3} (\cos (2 t) (\frac{1}{3} e^{3 t}) - (\frac{1}{3} \cdot - 2 \int e^{3 t} (\sin (2 t)) d t))))$

Step 6.8

Simplify terms.

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Step 6.8.1

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

$e^{3 t} y = 13 (\frac{\sin (2 t) e^{3 t}}{3} - (\frac{2}{3} (\cos (2 t) \frac{e^{3 t}}{3} - (\frac{1}{3} \cdot - 2 \int e^{3 t} (\sin (2 t)) d t))))$

Step 6.8.2

Combine $\cos (2 t)$ and $\frac{e^{3 t}}{3}$ .

$e^{3 t} y = 13 (\frac{\sin (2 t) e^{3 t}}{3} - (\frac{2}{3} (\frac{\cos (2 t) e^{3 t}}{3} - (\frac{1}{3} \cdot - 2 \int e^{3 t} (\sin (2 t)) d t))))$

Step 6.8.3

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

$e^{3 t} y = 13 (\frac{\sin (2 t) e^{3 t}}{3} - (\frac{2}{3} (\frac{\cos (2 t) e^{3 t}}{3} - (\frac{- 2}{3} \int e^{3 t} (\sin (2 t)) d t))))$

Step 6.8.4

Apply the distributive property.

$e^{3 t} y = 13 (\frac{\sin (2 t) e^{3 t}}{3} - \frac{2}{3} \cdot \frac{\cos (2 t) e^{3 t}}{3} - \frac{2}{3} (- (\frac{- 2}{3} \int e^{3 t} (\sin (2 t)) d t)))$

Step 6.8.5

Multiply $\frac{\cos (2 t) e^{3 t}}{3}$ by $\frac{2}{3}$ .

$e^{3 t} y = 13 (\frac{\sin (2 t) e^{3 t}}{3} - \frac{\cos (2 t) e^{3 t} \cdot 2}{3 \cdot 3} - \frac{2}{3} (- (\frac{- 2}{3} \int e^{3 t} (\sin (2 t)) d t)))$

Step 6.8.6

Multiply.

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Step 6.8.6.1

Multiply $3$ by $3$ .

$e^{3 t} y = 13 (\frac{\sin (2 t) e^{3 t}}{3} - \frac{\cos (2 t) e^{3 t} \cdot 2}{9} - \frac{2}{3} (- (\frac{- 2}{3} \int e^{3 t} (\sin (2 t)) d t)))$

Step 6.8.6.2

Multiply $- 1$ by $- 1$ .

$e^{3 t} y = 13 (\frac{\sin (2 t) e^{3 t}}{3} - \frac{\cos (2 t) e^{3 t} \cdot 2}{9} + 1 (\frac{2}{3}) (\frac{- 2}{3} \int e^{3 t} (\sin (2 t)) d t))$

Step 6.8.6.3

Multiply $\frac{2}{3}$ by $1$ .

$e^{3 t} y = 13 (\frac{\sin (2 t) e^{3 t}}{3} - \frac{\cos (2 t) e^{3 t} \cdot 2}{9} + \frac{2}{3} (\frac{- 2}{3} \int e^{3 t} (\sin (2 t)) d t))$

Step 6.8.7

Multiply $\frac{- 2}{3}$ by $\frac{2}{3}$ .

$e^{3 t} y = 13 (\frac{\sin (2 t) e^{3 t}}{3} - \frac{\cos (2 t) e^{3 t} \cdot 2}{9} + \frac{- 2 \cdot 2}{3 \cdot 3} \int e^{3 t} (\sin (2 t)) d t)$

Step 6.8.8

Multiply.

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Step 6.8.8.1

Multiply $- 2$ by $2$ .

$e^{3 t} y = 13 (\frac{\sin (2 t) e^{3 t}}{3} - \frac{\cos (2 t) e^{3 t} \cdot 2}{9} + \frac{- 4}{3 \cdot 3} \int e^{3 t} (\sin (2 t)) d t)$

Step 6.8.8.2

Multiply $3$ by $3$ .

$e^{3 t} y = 13 (\frac{\sin (2 t) e^{3 t}}{3} - \frac{\cos (2 t) e^{3 t} \cdot 2}{9} + \frac{- 4}{9} \int e^{3 t} (\sin (2 t)) d t)$

Step 6.9

Solving for $\int e^{3 t} \sin (2 t) d t$ , we find that $\int e^{3 t} \sin (2 t) d t$ = $\frac{\frac{\sin (2 t) e^{3 t}}{3} - \frac{\cos (2 t) e^{3 t} \cdot 2}{9}}{\frac{13}{9}}$ .

$e^{3 t} y = 13 (\frac{\frac{\sin (2 t) e^{3 t}}{3} - \frac{\cos (2 t) e^{3 t} \cdot 2}{9}}{\frac{13}{9}} + C)$

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 $2$ to the left of $\cos (2 t) e^{3 t}$ .

$e^{3 t} y = 13 (\frac{\frac{\sin (2 t) e^{3 t}}{3} - \frac{2 \cdot (\cos (2 t) e^{3 t})}{9}}{\frac{13}{9}} + C)$

Step 6.10.1.2

Multiply by the reciprocal of the fraction to divide by $\frac{13}{9}$ .

$e^{3 t} y = 13 ((\frac{\sin (2 t) e^{3 t}}{3} - \frac{2 \cos (2 t) e^{3 t}}{9}) \frac{9}{13} + C)$

Step 6.10.2

Rewrite $13 ((\frac{\sin (2 t) e^{3 t}}{3} - \frac{2 \cos (2 t) e^{3 t}}{9}) \frac{9}{13} + C)$ as $13 (\frac{\sin (2 t) e^{3 t} \cdot 9}{3 \cdot 13} + \frac{- 2 \cos (2 t) e^{3 t}}{13}) + C$ .

$e^{3 t} y = 13 (\frac{\sin (2 t) e^{3 t} \cdot 9}{3 \cdot 13} + \frac{- 2 \cos (2 t) e^{3 t}}{13}) + C$

Step 6.10.3

Simplify.

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Step 6.10.3.1

Move $9$ to the left of $\sin (2 t) e^{3 t}$ .

$e^{3 t} y = 13 (\frac{9 \cdot (\sin (2 t) e^{3 t})}{3 \cdot 13} + \frac{- 2 \cos (2 t) e^{3 t}}{13}) + C$

Step 6.10.3.2

Multiply $3$ by $13$ .

$e^{3 t} y = 13 (\frac{9 \sin (2 t) e^{3 t}}{39} + \frac{- 2 \cos (2 t) e^{3 t}}{13}) + C$

Step 6.10.3.3

Cancel the common factor of $9$ and $39$ .

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Step 6.10.3.3.1

Factor $3$ out of $9 \sin (2 t) e^{3 t}$ .

$e^{3 t} y = 13 (\frac{3 (3 \sin (2 t) e^{3 t})}{39} + \frac{- 2 \cos (2 t) e^{3 t}}{13}) + C$

Step 6.10.3.3.2

Cancel the common factors.

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Step 6.10.3.3.2.1

Factor $3$ out of $39$ .

$e^{3 t} y = 13 (\frac{3 (3 \sin (2 t) e^{3 t})}{3 (13)} + \frac{- 2 \cos (2 t) e^{3 t}}{13}) + C$

Step 6.10.3.3.2.2

Cancel the common factor.

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

Step 6.10.3.3.2.3

Rewrite the expression.

$e^{3 t} y = 13 (\frac{3 \sin (2 t) e^{3 t}}{13} + \frac{- 2 \cos (2 t) e^{3 t}}{13}) + C$

Step 6.10.3.4

Move the negative in front of the fraction.

$e^{3 t} y = 13 (\frac{3 \sin (2 t) e^{3 t}}{13} - \frac{2 \cos (2 t) e^{3 t}}{13}) + C$

Step 6.10.4

Simplify.

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Step 6.10.4.1

Reorder factors in $\frac{3 \sin (2 t) e^{3 t}}{13}$ .

$e^{3 t} y = 13 (\frac{3 e^{3 t} \sin (2 t)}{13} - \frac{2 \cos (2 t) e^{3 t}}{13}) + C$

Step 6.10.4.2

Reorder factors in $\frac{2 \cos (2 t) e^{3 t}}{13}$ .

$e^{3 t} y = 13 (\frac{3}{13} e^{3 t} \sin (2 t) - \frac{2}{13} e^{3 t} \cos (2 t)) + C$

Step 7

Solve for $y$ .

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Step 7.1

Simplify.

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Step 7.1.1

Simplify each term.

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Step 7.1.1.1

Multiply $\frac{3}{13} (e^{3 t} \sin (2 t))$ .

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Step 7.1.1.1.1

Combine $e^{3 t}$ and $\frac{3}{13}$ .

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

Step 7.1.1.1.2

Combine $\frac{e^{3 t} \cdot 3}{13}$ and $\sin (2 t)$ .

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

Step 7.1.1.2

Move $3$ to the left of $e^{3 t}$ .

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

Step 7.1.1.3

Multiply $- \frac{2}{13} (e^{3 t} \cos (2 t))$ .

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Step 7.1.1.3.1

Combine $e^{3 t}$ and $\frac{2}{13}$ .

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

Step 7.1.1.3.2

Combine $\cos (2 t)$ and $\frac{e^{3 t} \cdot 2}{13}$ .

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

Step 7.1.1.4

Remove unnecessary parentheses.

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

Step 7.1.1.5

Move $2$ to the left of $\cos (2 t) e^{3 t}$ .

$e^{3 t} y = 13 (\frac{3 e^{3 t} \sin (2 t)}{13} - \frac{2 \cos (2 t) e^{3 t}}{13}) + C$

Step 7.1.2

Apply the distributive property.

$e^{3 t} y = 13 \frac{3 e^{3 t} \sin (2 t)}{13} + 13 (- \frac{2 \cos (2 t) e^{3 t}}{13}) + C$

Step 7.1.3

Cancel the common factor of $13$ .

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Step 7.1.3.1

Cancel the common factor.

$e^{3 t} y = 13 \frac{3 e^{3 t} \sin (2 t)}{13} + 13 (- \frac{2 \cos (2 t) e^{3 t}}{13}) + C$

Step 7.1.3.2

Rewrite the expression.

$e^{3 t} y = 3 e^{3 t} \sin (2 t) + 13 (- \frac{2 \cos (2 t) e^{3 t}}{13}) + C$

Step 7.1.4

Cancel the common factor of $13$ .

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Step 7.1.4.1

Move the leading negative in $- \frac{2 \cos (2 t) e^{3 t}}{13}$ into the numerator.

$e^{3 t} y = 3 e^{3 t} \sin (2 t) + 13 \frac{- 2 \cos (2 t) e^{3 t}}{13} + C$

Step 7.1.4.2

Cancel the common factor.

$e^{3 t} y = 3 e^{3 t} \sin (2 t) + 13 \frac{- 2 \cos (2 t) e^{3 t}}{13} + C$

Step 7.1.4.3

Rewrite the expression.

$e^{3 t} y = 3 e^{3 t} \sin (2 t) - 2 \cos (2 t) e^{3 t} + C$

Step 7.1.5

Reorder factors in $3 e^{3 t} \sin (2 t) - 2 \cos (2 t) e^{3 t}$ .

$e^{3 t} y = 3 e^{3 t} \sin (2 t) - 2 e^{3 t} \cos (2 t) + C$

Step 7.2

Divide each term in $e^{3 t} y = 3 e^{3 t} \sin (2 t) - 2 e^{3 t} \cos (2 t) + C$ by $e^{3 t}$ and simplify.

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Step 7.2.1

Divide each term in $e^{3 t} y = 3 e^{3 t} \sin (2 t) - 2 e^{3 t} \cos (2 t) + C$ by $e^{3 t}$ .

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

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

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

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Step 7.2.2.1.1

Cancel the common factor.

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

Step 7.2.2.1.2

Divide $y$ by $1$ .

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

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

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

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Step 7.2.3.1.1.1

Cancel the common factor.

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

Step 7.2.3.1.1.2

Divide $3 \sin (2 t)$ by $1$ .

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

Step 7.2.3.1.2

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

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Step 7.2.3.1.2.1

Cancel the common factor.

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

Step 7.2.3.1.2.2

Divide $- 2 \cos (2 t)$ by $1$ .

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

Step 8

Use the initial condition to find the value of $C$ by substituting $0$ for $t$ and $6$ for $y$ in $y = 3 \sin (2 t) - 2 \cos (2 t) + \frac{C}{e^{3 t}}$ .

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

Step 9

Solve for $C$ .

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Step 9.1

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

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

Step 9.2

Simplify the left side.

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Step 9.2.1

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

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Step 9.2.1.1

Simplify each term.

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Step 9.2.1.1.1

Multiply $2$ by $0$ .

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

Step 9.2.1.1.2

The exact value of $\sin (0)$ is $0$ .

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

Step 9.2.1.1.3

Multiply $3$ by $0$ .

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

Step 9.2.1.1.4

Multiply $2$ by $0$ .

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

Step 9.2.1.1.5

The exact value of $\cos (0)$ is $1$ .

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

Step 9.2.1.1.6

Multiply $- 2$ by $1$ .

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

Step 9.2.1.1.7

Simplify the denominator.

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Step 9.2.1.1.7.1

Multiply $3$ by $0$ .

$0 - 2 + \frac{C}{e^{0}} = 6$

Step 9.2.1.1.7.2

Anything raised to $0$ is $1$ .

$0 - 2 + \frac{C}{1} = 6$

Step 9.2.1.1.8

Divide $C$ by $1$ .

$0 - 2 + C = 6$

Step 9.2.1.2

Subtract $2$ from $0$ .

$- 2 + C = 6$

Step 9.3

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

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Step 9.3.1

Add $2$ to both sides of the equation.

$C = 6 + 2$

Step 9.3.2

Add $6$ and $2$ .

$C = 8$

Step 10

Substitute $8$ for $C$ in $y = 3 \sin (2 t) - 2 \cos (2 t) + \frac{C}{e^{3 t}}$ and simplify.

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Step 10.1

Substitute $8$ for $C$ .

$y = 3 \sin (2 t) - 2 \cos (2 t) + \frac{8}{e^{3 t}}$