Calculus Examples

$2 \frac{d y}{d t} - y = 4 \sin (3 t)$

Step 1

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

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Step 1.1

Divide each term in $2 \frac{d y}{d t} - y = 4 \sin (3 t)$ by $2$ .

$\frac{2 \frac{d y}{d t}}{2} + \frac{- y}{2} = \frac{4 \sin (3 t)}{2}$

Step 1.2

Cancel the common factor of $2$ .

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Step 1.2.1

Cancel the common factor.

$\frac{2 \frac{d y}{d t}}{2} + \frac{- y}{2} = \frac{4 \sin (3 t)}{2}$

Step 1.2.2

Divide $\frac{d y}{d t}$ by $1$ .

$\frac{d y}{d t} + \frac{- y}{2} = \frac{4 \sin (3 t)}{2}$

Step 1.3

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

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Step 1.3.1

Factor $2$ out of $4 \sin (3 t)$ .

$\frac{d y}{d t} + \frac{- y}{2} = \frac{2 (2 \sin (3 t))}{2}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $2$ out of $2$ .

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

Step 1.3.2.2

Cancel the common factor.

$\frac{d y}{d t} + \frac{- y}{2} = \frac{2 (2 \sin (3 t))}{2 \cdot 1}$

Step 1.3.2.3

Rewrite the expression.

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

Step 1.3.2.4

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

$\frac{d y}{d t} + \frac{- y}{2} = 2 \sin (3 t)$

Step 1.4

Reorder terms.

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

Step 2

The integrating factor is defined by the formula $e^{\int P (t) d t}$ , where $P (t) = \frac{- 1}{2}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{- 1}{2} d t}$

Step 2.2

Integrate $\frac{- 1}{2}$ .

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Step 2.2.1

Move the negative in front of the fraction.

$e^{\int - \frac{1}{2} d t}$

Step 2.2.2

Apply the constant rule.

$e^{- \frac{1}{2} t + C}$

Step 2.3

Remove the constant of integration.

$e^{- \frac{1}{2} t}$

Step 2.4

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

$e^{- \frac{t}{2}}$

Step 3

Multiply each term by the integrating factor $e^{- \frac{t}{2}}$ .

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Step 3.1

Multiply each term by $e^{- \frac{t}{2}}$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 3.2.3

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

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

Step 3.2.4

Combine $y$ and $\frac{e^{- \frac{t}{2}}}{2}$ .

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

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

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

Step 7.2

Let $u = - \frac{t}{2}$ . Then $d u = - \frac{1}{2} d t$ , so $- 2 d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.2.1

Let $u = - \frac{t}{2}$ . Find $\frac{d u}{d t}$ .

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Step 7.2.1.1

Differentiate $- \frac{t}{2}$ .

$\frac{d}{d t} [- \frac{t}{2}]$

Step 7.2.1.2

Since $- \frac{1}{2}$ is constant with respect to $t$ , the derivative of $- \frac{t}{2}$ with respect to $t$ is $- \frac{1}{2} \frac{d}{d t} [t]$ .

$- \frac{1}{2} \frac{d}{d t} [t]$

Step 7.2.1.3

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

$- \frac{1}{2} \cdot 1$

Step 7.2.1.4

Multiply $- 1$ by $1$ .

$- \frac{1}{2}$

Step 7.2.2

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

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

Step 7.3

Simplify.

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Step 7.3.1

Multiply $- 2$ by $3$ .

$e^{- \frac{t}{2}} y = 2 \int - e^{u} \sin (- 6 u) \frac{- 1}{- \frac{1}{2}} d u$

Step 7.3.2

Dividing two negative values results in a positive value.

$e^{- \frac{t}{2}} y = 2 \int - e^{u} \sin (- 6 u) \frac{1}{\frac{1}{2}} d u$

Step 7.3.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$e^{- \frac{t}{2}} y = 2 \int - e^{u} \sin (- 6 u) (1 \cdot 2) d u$

Step 7.3.4

Multiply $2$ by $1$ .

$e^{- \frac{t}{2}} y = 2 \int - e^{u} \sin (- 6 u) \cdot 2 d u$

Step 7.3.5

Multiply $2$ by $- 1$ .

$e^{- \frac{t}{2}} y = 2 \int - 2 e^{u} \sin (- 6 u) d u$

Step 7.4

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

$e^{- \frac{t}{2}} y = 2 (- 2 \int e^{u} \sin (- 6 u) d u)$

Step 7.5

Simplify the expression.

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Step 7.5.1

Multiply $- 2$ by $2$ .

$e^{- \frac{t}{2}} y = - 4 \int e^{u} \sin (- 6 u) d u$

Step 7.5.2

Reorder $e^{u}$ and $\sin (- 6 u)$ .

$e^{- \frac{t}{2}} y = - 4 \int \sin (- 6 u) e^{u} d u$

Step 7.6

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

$e^{- \frac{t}{2}} y = - 4 (\sin (- 6 u) e^{u} - \int e^{u} (- 6 \cos (- 6 u)) d u)$

Step 7.7

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

$e^{- \frac{t}{2}} y = - 4 (\sin (- 6 u) e^{u} - (- 6 \int e^{u} (\cos (- 6 u)) d u))$

Step 7.8

Simplify the expression.

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Step 7.8.1

Multiply $- 6$ by $- 1$ .

$e^{- \frac{t}{2}} y = - 4 (\sin (- 6 u) e^{u} + 6 \int e^{u} (\cos (- 6 u)) d u)$

Step 7.8.2

Reorder $e^{u}$ and $\cos (- 6 u)$ .

$e^{- \frac{t}{2}} y = - 4 (\sin (- 6 u) e^{u} + 6 \int \cos (- 6 u) e^{u} d u)$

Step 7.9

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

$e^{- \frac{t}{2}} y = - 4 (\sin (- 6 u) e^{u} + 6 (\cos (- 6 u) e^{u} - \int e^{u} (6 \sin (- 6 u)) d u))$

Step 7.10

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

$e^{- \frac{t}{2}} y = - 4 (\sin (- 6 u) e^{u} + 6 (\cos (- 6 u) e^{u} - (6 \int e^{u} (\sin (- 6 u)) d u)))$

Step 7.11

Simplify by multiplying through.

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Step 7.11.1

Multiply $6$ by $- 1$ .

$e^{- \frac{t}{2}} y = - 4 (\sin (- 6 u) e^{u} + 6 (\cos (- 6 u) e^{u} - 6 \int e^{u} (\sin (- 6 u)) d u))$

Step 7.11.2

Apply the distributive property.

$e^{- \frac{t}{2}} y = - 4 (\sin (- 6 u) e^{u} + 6 (\cos (- 6 u) e^{u}) + 6 (- 6 \int e^{u} (\sin (- 6 u)) d u))$

Step 7.11.3

Multiply $- 6$ by $6$ .

$e^{- \frac{t}{2}} y = - 4 (\sin (- 6 u) e^{u} + 6 (\cos (- 6 u) e^{u}) - 36 \int e^{u} (\sin (- 6 u)) d u)$

Step 7.12

Solving for $\int e^{u} \sin (- 6 u) d u$ , we find that $\int e^{u} \sin (- 6 u) d u$ = $\frac{\sin (- 6 u) e^{u} + 6 (\cos (- 6 u) e^{u})}{37}$ .

$e^{- \frac{t}{2}} y = - 4 (\frac{\sin (- 6 u) e^{u} + 6 (\cos (- 6 u) e^{u})}{37} + C)$

Step 7.13

Rewrite $- 4 (\frac{\sin (- 6 u) e^{u} + 6 \cos (- 6 u) e^{u}}{37} + C)$ as $- 4 \frac{e^{u} (\sin (- 6 u) + 6 \cos (- 6 u))}{37} + C$ .

$e^{- \frac{t}{2}} y = - 4 \frac{e^{u} (\sin (- 6 u) + 6 \cos (- 6 u))}{37} + C$

Step 7.14

Simplify.

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Step 7.14.1

Combine $- 4$ and $\frac{e^{u} (\sin (- 6 u) + 6 \cos (- 6 u))}{37}$ .

$e^{- \frac{t}{2}} y = \frac{- 4 (e^{u} (\sin (- 6 u) + 6 \cos (- 6 u)))}{37} + C$

Step 7.14.2

Move the negative in front of the fraction.

$e^{- \frac{t}{2}} y = - \frac{4 e^{u} (\sin (- 6 u) + 6 \cos (- 6 u))}{37} + C$

Step 7.15

Replace all occurrences of $u$ with $- \frac{t}{2}$ .

$e^{- \frac{t}{2}} y = - \frac{4 e^{- \frac{t}{2}} (\sin (- 6 (- \frac{t}{2})) + 6 \cos (- 6 (- \frac{t}{2})))}{37} + C$

Step 7.16

Simplify.

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Step 7.16.1

Multiply $- 1$ by $- 6$ .

$e^{- \frac{t}{2}} y = - \frac{4 e^{- \frac{t}{2}} (\sin (6 \frac{t}{2}) + 6 \cos (- 6 (- \frac{t}{2})))}{37} + C$

Step 7.16.2

Combine $6$ and $\frac{t}{2}$ .

$e^{- \frac{t}{2}} y = - \frac{4 e^{- \frac{t}{2}} (\sin (\frac{6 t}{2}) + 6 \cos (- 6 (- \frac{t}{2})))}{37} + C$

Step 7.16.3

Multiply $- 1$ by $- 6$ .

$e^{- \frac{t}{2}} y = - \frac{4 e^{- \frac{t}{2}} (\sin (\frac{6 t}{2}) + 6 \cos (6 \frac{t}{2}))}{37} + C$

Step 7.16.4

Combine $6$ and $\frac{t}{2}$ .

$e^{- \frac{t}{2}} y = - \frac{4 e^{- \frac{t}{2}} (\sin (\frac{6 t}{2}) + 6 \cos (\frac{6 t}{2}))}{37} + C$

Step 7.17

Simplify.

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Step 7.17.1

Cancel the common factor of $6$ and $2$ .

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Step 7.17.1.1

Factor $2$ out of $6 t$ .

$e^{- \frac{t}{2}} y = - \frac{4 e^{- \frac{t}{2}} (\sin (\frac{2 (3 t)}{2}) + 6 \cos (\frac{6 t}{2}))}{37} + C$

Step 7.17.1.2

Cancel the common factors.

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Step 7.17.1.2.1

Factor $2$ out of $2$ .

$e^{- \frac{t}{2}} y = - \frac{4 e^{- \frac{t}{2}} (\sin (\frac{2 (3 t)}{2 (1)}) + 6 \cos (\frac{6 t}{2}))}{37} + C$

Step 7.17.1.2.2

Cancel the common factor.

$e^{- \frac{t}{2}} y = - \frac{4 e^{- \frac{t}{2}} (\sin (\frac{2 (3 t)}{2 \cdot 1}) + 6 \cos (\frac{6 t}{2}))}{37} + C$

Step 7.17.1.2.3

Rewrite the expression.

$e^{- \frac{t}{2}} y = - \frac{4 e^{- \frac{t}{2}} (\sin (\frac{3 t}{1}) + 6 \cos (\frac{6 t}{2}))}{37} + C$

Step 7.17.1.2.4

Divide $3 t$ by $1$ .

$e^{- \frac{t}{2}} y = - \frac{4 e^{- \frac{t}{2}} (\sin (3 t) + 6 \cos (\frac{6 t}{2}))}{37} + C$

Step 7.17.2

Cancel the common factor of $6$ and $2$ .

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Step 7.17.2.1

Factor $2$ out of $6 t$ .

$e^{- \frac{t}{2}} y = - \frac{4 e^{- \frac{t}{2}} (\sin (3 t) + 6 \cos (\frac{2 (3 t)}{2}))}{37} + C$

Step 7.17.2.2

Cancel the common factors.

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Step 7.17.2.2.1

Factor $2$ out of $2$ .

$e^{- \frac{t}{2}} y = - \frac{4 e^{- \frac{t}{2}} (\sin (3 t) + 6 \cos (\frac{2 (3 t)}{2 (1)}))}{37} + C$

Step 7.17.2.2.2

Cancel the common factor.

$e^{- \frac{t}{2}} y = - \frac{4 e^{- \frac{t}{2}} (\sin (3 t) + 6 \cos (\frac{2 (3 t)}{2 \cdot 1}))}{37} + C$

Step 7.17.2.2.3

Rewrite the expression.

$e^{- \frac{t}{2}} y = - \frac{4 e^{- \frac{t}{2}} (\sin (3 t) + 6 \cos (\frac{3 t}{1}))}{37} + C$

Step 7.17.2.2.4

Divide $3 t$ by $1$ .

$e^{- \frac{t}{2}} y = - \frac{4 e^{- \frac{t}{2}} (\sin (3 t) + 6 \cos (3 t))}{37} + C$

$e^{- \frac{t}{2}} y = - \frac{4}{37} e^{- \frac{1}{2} t} (\sin (3 t) + 6 \cos (3 t)) + C$

Step 8

Solve for $y$ .

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Step 8.1

Simplify.

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Step 8.1.1

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

$e^{- \frac{t}{2}} y = - \frac{4}{37} \cdot (e^{- \frac{t}{2}} (\sin (3 t) + 6 \cos (3 t))) + C$

Step 8.1.2

Combine $e^{- \frac{t}{2}}$ and $\frac{4}{37}$ .

$e^{- \frac{t}{2}} y = - \frac{e^{- \frac{t}{2}} \cdot 4}{37} \cdot (\sin (3 t) + 6 \cos (3 t)) + C$

Step 8.2

Divide each term in $e^{- \frac{t}{2}} y = - \frac{e^{- \frac{t}{2}} \cdot 4}{37} \cdot (\sin (3 t) + 6 \cos (3 t)) + C$ by $e^{- \frac{t}{2}}$ and simplify.

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Step 8.2.1

Divide each term in $e^{- \frac{t}{2}} y = - \frac{e^{- \frac{t}{2}} \cdot 4}{37} \cdot (\sin (3 t) + 6 \cos (3 t)) + C$ by $e^{- \frac{t}{2}}$ .

$\frac{e^{- \frac{t}{2}} y}{e^{- \frac{t}{2}}} = \frac{- \frac{e^{- \frac{t}{2}} \cdot 4}{37} \cdot (\sin (3 t) + 6 \cos (3 t))}{e^{- \frac{t}{2}}} + \frac{C}{e^{- \frac{t}{2}}}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $e^{- \frac{t}{2}}$ .

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{e^{- \frac{t}{2}} y}{e^{- \frac{t}{2}}} = \frac{- \frac{e^{- \frac{t}{2}} \cdot 4}{37} \cdot (\sin (3 t) + 6 \cos (3 t))}{e^{- \frac{t}{2}}} + \frac{C}{e^{- \frac{t}{2}}}$

Step 8.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{e^{- \frac{t}{2}} \cdot 4}{37} \cdot (\sin (3 t) + 6 \cos (3 t))}{e^{- \frac{t}{2}}} + \frac{C}{e^{- \frac{t}{2}}}$

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

Move $4$ to the left of $e^{- \frac{t}{2}}$ .

$y = \frac{- \frac{4 e^{- \frac{t}{2}}}{37} (\sin (3 t) + 6 \cos (3 t))}{e^{- \frac{t}{2}}} + \frac{C}{e^{- \frac{t}{2}}}$

Step 8.2.3.1.2

Factor $\frac{4 e^{- \frac{t}{2}}}{37}$ out of $\frac{- \frac{4 e^{- \frac{t}{2}}}{37} (\sin (3 t) + 6 \cos (3 t))}{e^{- \frac{t}{2}}}$ .

$y = \frac{4 e^{- \frac{t}{2}}}{37} \cdot \frac{- 1 (\sin (3 t) + 6 \cos (3 t))}{e^{- \frac{t}{2}}} + \frac{C}{e^{- \frac{t}{2}}}$

Step 8.2.3.1.3

Combine.

$y = \frac{4 e^{- \frac{t}{2}} (- 1 (\sin (3 t) + 6 \cos (3 t)))}{37 e^{- \frac{t}{2}}} + \frac{C}{e^{- \frac{t}{2}}}$

Step 8.2.3.1.4

Cancel the common factor of $e^{- \frac{t}{2}}$ .

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Step 8.2.3.1.4.1

Cancel the common factor.

$y = \frac{4 e^{- \frac{t}{2}} (- 1 (\sin (3 t) + 6 \cos (3 t)))}{37 e^{- \frac{t}{2}}} + \frac{C}{e^{- \frac{t}{2}}}$

Step 8.2.3.1.4.2

Rewrite the expression.

$y = \frac{4 (- 1 (\sin (3 t) + 6 \cos (3 t)))}{37} + \frac{C}{e^{- \frac{t}{2}}}$

Step 8.2.3.1.5

Multiply $- 1$ by $4$ .

$y = \frac{- 4 (\sin (3 t) + 6 \cos (3 t))}{37} + \frac{C}{e^{- \frac{t}{2}}}$

Step 8.2.3.1.6

Move the negative in front of the fraction.

$y = - \frac{4 (\sin (3 t) + 6 \cos (3 t))}{37} + \frac{C}{e^{- \frac{t}{2}}}$