Calculus Examples

$25 \frac{d T}{d t} + T = 80 e^{- 0.04 t}$

Step 1

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

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Step 1.1

Divide each term in $25 \frac{d T}{d t} + T = 80 e^{- 0.04 t}$ by $25$ .

$\frac{25 \frac{d T}{d t}}{25} + \frac{T}{25} = \frac{80 e^{- 0.04 t}}{25}$

Step 1.2

Cancel the common factor of $25$ .

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Step 1.2.1

Cancel the common factor.

$\frac{25 \frac{d T}{d t}}{25} + \frac{T}{25} = \frac{80 e^{- 0.04 t}}{25}$

Step 1.2.2

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

$\frac{d T}{d t} + \frac{T}{25} = \frac{80 e^{- 0.04 t}}{25}$

Step 1.3

Cancel the common factor of $80$ and $25$ .

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Step 1.3.1

Factor $5$ out of $80 e^{- 0.04 t}$ .

$\frac{d T}{d t} + \frac{T}{25} = \frac{5 (16 e^{- 0.04 t})}{25}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $5$ out of $25$ .

$\frac{d T}{d t} + \frac{T}{25} = \frac{5 (16 e^{- 0.04 t})}{5 (5)}$

Step 1.3.2.2

Cancel the common factor.

$\frac{d T}{d t} + \frac{T}{25} = \frac{5 (16 e^{- 0.04 t})}{5 \cdot 5}$

Step 1.3.2.3

Rewrite the expression.

$\frac{d T}{d t} + \frac{T}{25} = \frac{16 e^{- 0.04 t}}{5}$

Step 1.4

Reorder terms.

$\frac{d T}{d t} + \frac{1}{25} T = \frac{16}{5} e^{- 0.04 t}$

Step 2

The integrating factor is defined by the formula $e^{\int P (t) d t}$ , where $P (t) = \frac{1}{25}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{1}{25} d t}$

Step 2.2

Apply the constant rule.

$e^{\frac{1}{25} t + C}$

Step 2.3

Remove the constant of integration.

$e^{\frac{1}{25} t}$

Step 2.4

Combine $\frac{1}{25}$ and $t$ .

$e^{\frac{t}{25}}$

Step 3

Multiply each term by the integrating factor $e^{\frac{t}{25}}$ .

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Step 3.1

Multiply each term by $e^{\frac{t}{25}}$ .

$e^{\frac{t}{25}} \frac{d T}{d t} + e^{\frac{t}{25}} (\frac{1}{25} T) = e^{\frac{t}{25}} (\frac{16}{5} e^{- 0.04 t})$

Step 3.2

Simplify each term.

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Step 3.2.1

Rewrite using the commutative property of multiplication.

$e^{\frac{t}{25}} \frac{d T}{d t} + \frac{1}{25} e^{\frac{t}{25}} T = e^{\frac{t}{25}} (\frac{16}{5} e^{- 0.04 t})$

Step 3.2.2

Combine $\frac{1}{25}$ and $e^{\frac{t}{25}}$ .

$e^{\frac{t}{25}} \frac{d T}{d t} + \frac{e^{\frac{t}{25}}}{25} T = e^{\frac{t}{25}} (\frac{16}{5} e^{- 0.04 t})$

Step 3.2.3

Combine $\frac{e^{\frac{t}{25}}}{25}$ and $T$ .

$e^{\frac{t}{25}} \frac{d T}{d t} + \frac{e^{\frac{t}{25}} T}{25} = e^{\frac{t}{25}} (\frac{16}{5} e^{- 0.04 t})$

Step 3.3

Rewrite using the commutative property of multiplication.

$e^{\frac{t}{25}} \frac{d T}{d t} + \frac{e^{\frac{t}{25}} T}{25} = \frac{16}{5} e^{\frac{t}{25}} e^{- 0.04 t}$

Step 3.4

Multiply $e^{\frac{t}{25}}$ by $e^{- 0.04 t}$ by adding the exponents.

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Step 3.4.1

Move $e^{- 0.04 t}$ .

$e^{\frac{t}{25}} \frac{d T}{d t} + \frac{e^{\frac{t}{25}} T}{25} = \frac{16}{5} (e^{- 0.04 t} e^{\frac{t}{25}})$

Step 3.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$e^{\frac{t}{25}} \frac{d T}{d t} + \frac{e^{\frac{t}{25}} T}{25} = \frac{16}{5} e^{- 0.04 t + \frac{t}{25}}$

Step 3.4.3

To write $- 0.04 t$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$e^{\frac{t}{25}} \frac{d T}{d t} + \frac{e^{\frac{t}{25}} T}{25} = \frac{16}{5} e^{- 0.04 t \cdot \frac{25}{25} + \frac{t}{25}}$

Step 3.4.4

Combine $- 0.04 t$ and $\frac{25}{25}$ .

$e^{\frac{t}{25}} \frac{d T}{d t} + \frac{e^{\frac{t}{25}} T}{25} = \frac{16}{5} e^{\frac{- 0.04 t \cdot 25}{25} + \frac{t}{25}}$

Step 3.4.5

Combine the numerators over the common denominator.

$e^{\frac{t}{25}} \frac{d T}{d t} + \frac{e^{\frac{t}{25}} T}{25} = \frac{16}{5} e^{\frac{- 0.04 t \cdot 25 + t}{25}}$

Step 3.4.6

Simplify the numerator.

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Step 3.4.6.1

Factor $t$ out of $- 0.04 t \cdot 25 + t$ .

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Step 3.4.6.1.1

Factor $t$ out of $- 0.04 t \cdot 25$ .

$e^{\frac{t}{25}} \frac{d T}{d t} + \frac{e^{\frac{t}{25}} T}{25} = \frac{16}{5} e^{\frac{t (- 0.04 \cdot 25) + t}{25}}$

Step 3.4.6.1.2

Raise $t$ to the power of $1$ .

$e^{\frac{t}{25}} \frac{d T}{d t} + \frac{e^{\frac{t}{25}} T}{25} = \frac{16}{5} e^{\frac{t (- 0.04 \cdot 25) + t^{1}}{25}}$

Step 3.4.6.1.3

Factor $t$ out of $t^{1}$ .

$e^{\frac{t}{25}} \frac{d T}{d t} + \frac{e^{\frac{t}{25}} T}{25} = \frac{16}{5} e^{\frac{t (- 0.04 \cdot 25) + t \cdot 1}{25}}$

Step 3.4.6.1.4

Factor $t$ out of $t (- 0.04 \cdot 25) + t \cdot 1$ .

$e^{\frac{t}{25}} \frac{d T}{d t} + \frac{e^{\frac{t}{25}} T}{25} = \frac{16}{5} e^{\frac{t (- 0.04 \cdot 25 + 1)}{25}}$

Step 3.4.6.2

Multiply $- 0.04$ by $25$ .

$e^{\frac{t}{25}} \frac{d T}{d t} + \frac{e^{\frac{t}{25}} T}{25} = \frac{16}{5} e^{\frac{t (- 1 + 1)}{25}}$

Step 3.4.6.3

Add $- 1$ and $1$ .

$e^{\frac{t}{25}} \frac{d T}{d t} + \frac{e^{\frac{t}{25}} T}{25} = \frac{16}{5} e^{\frac{t \cdot 0}{25}}$

Step 3.4.7

Multiply $t$ by $0$ .

$e^{\frac{t}{25}} \frac{d T}{d t} + \frac{e^{\frac{t}{25}} T}{25} = \frac{16}{5} e^{\frac{0}{25}}$

Step 3.4.8

Divide $0$ by $25$ .

$e^{\frac{t}{25}} \frac{d T}{d t} + \frac{e^{\frac{t}{25}} T}{25} = \frac{16}{5} e^{0}$

Step 3.5

Simplify $\frac{16}{5} e^{0}$ .

$e^{\frac{t}{25}} \frac{d T}{d t} + \frac{e^{\frac{t}{25}} T}{25} = \frac{16}{5}$

Step 3.6

Reorder factors in $e^{\frac{t}{25}} \frac{d T}{d t} + \frac{e^{\frac{t}{25}} T}{25} = \frac{16}{5}$ .

$e^{\frac{t}{25}} \frac{d T}{d t} + \frac{T e^{\frac{t}{25}}}{25} = \frac{16}{5}$

Step 4

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

$\frac{d}{d t} [e^{\frac{t}{25}} T] = \frac{16}{5}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d t} [e^{\frac{t}{25}} T] d t = \int \frac{16}{5} d t$

Step 6

Integrate the left side.

$e^{\frac{t}{25}} T = \int \frac{16}{5} d t$

Step 7

Apply the constant rule.

$e^{\frac{t}{25}} T = \frac{16}{5} t + C$

Step 8

Divide each term in $e^{\frac{t}{25}} T = \frac{16}{5} t + C$ by $e^{\frac{t}{25}}$ and simplify.

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Step 8.1

Divide each term in $e^{\frac{t}{25}} T = \frac{16}{5} t + C$ by $e^{\frac{t}{25}}$ .

$\frac{e^{\frac{t}{25}} T}{e^{\frac{t}{25}}} = \frac{\frac{16}{5} t}{e^{\frac{t}{25}}} + \frac{C}{e^{\frac{t}{25}}}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $e^{\frac{t}{25}}$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{e^{\frac{t}{25}} T}{e^{\frac{t}{25}}} = \frac{\frac{16}{5} t}{e^{\frac{t}{25}}} + \frac{C}{e^{\frac{t}{25}}}$

Step 8.2.1.2

Divide $T$ by $1$ .

$T = \frac{\frac{16}{5} t}{e^{\frac{t}{25}}} + \frac{C}{e^{\frac{t}{25}}}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

Combine $\frac{16}{5}$ and $t$ .

$T = \frac{\frac{16 t}{5}}{e^{\frac{t}{25}}} + \frac{C}{e^{\frac{t}{25}}}$

Step 8.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$T = \frac{16 t}{5} \cdot \frac{1}{e^{\frac{t}{25}}} + \frac{C}{e^{\frac{t}{25}}}$

Step 8.3.1.3

Combine.

$T = \frac{16 t \cdot 1}{5 e^{\frac{t}{25}}} + \frac{C}{e^{\frac{t}{25}}}$

Step 8.3.1.4

Multiply $16$ by $1$ .

$T = \frac{16 t}{5 e^{\frac{t}{25}}} + \frac{C}{e^{\frac{t}{25}}}$