Calculus Examples

$\frac{d w}{d t} = 10 e^{\frac{- t}{25}} - \frac{w}{25}$

Step 1

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

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Step 1.1

Add $\frac{w}{25}$ to both sides of the equation.

$\frac{d w}{d t} + \frac{w}{25} = 10 e^{\frac{- t}{25}}$

Step 1.2

Rewrite the equation with isolated coefficients.

$\frac{d w}{d t} + \frac{1}{25} w = 10 e^{\frac{- 1}{25} t}$

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

$e^{\frac{t}{25}} \frac{d w}{d t} + e^{\frac{t}{25}} (\frac{1}{25} w) = e^{\frac{t}{25}} (10 e^{\frac{- 1}{25} t})$

Step 3.2

Simplify each term.

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Step 3.2.1

Rewrite using the commutative property of multiplication.

$e^{\frac{t}{25}} \frac{d w}{d t} + \frac{1}{25} e^{\frac{t}{25}} w = e^{\frac{t}{25}} (10 e^{\frac{- 1}{25} t})$

Step 3.2.2

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

$e^{\frac{t}{25}} \frac{d w}{d t} + \frac{e^{\frac{t}{25}}}{25} w = e^{\frac{t}{25}} (10 e^{\frac{- 1}{25} t})$

Step 3.2.3

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

$e^{\frac{t}{25}} \frac{d w}{d t} + \frac{e^{\frac{t}{25}} w}{25} = e^{\frac{t}{25}} (10 e^{\frac{- 1}{25} t})$

Step 3.3

Rewrite using the commutative property of multiplication.

$e^{\frac{t}{25}} \frac{d w}{d t} + \frac{e^{\frac{t}{25}} w}{25} = 10 e^{\frac{t}{25}} e^{\frac{- 1}{25} t}$

Step 3.4

Multiply $e^{\frac{t}{25}}$ by $e^{\frac{- 1}{25} t}$ by adding the exponents.

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Step 3.4.1

Move $e^{\frac{- 1}{25} t}$ .

$e^{\frac{t}{25}} \frac{d w}{d t} + \frac{e^{\frac{t}{25}} w}{25} = 10 (e^{\frac{- 1}{25} t} e^{\frac{t}{25}})$

Step 3.4.2

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

$e^{\frac{t}{25}} \frac{d w}{d t} + \frac{e^{\frac{t}{25}} w}{25} = 10 e^{\frac{- 1}{25} t + \frac{t}{25}}$

Step 3.4.3

Simplify each term.

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Step 3.4.3.1

Move the negative in front of the fraction.

$e^{\frac{t}{25}} \frac{d w}{d t} + \frac{e^{\frac{t}{25}} w}{25} = 10 e^{- \frac{1}{25} t + \frac{t}{25}}$

Step 3.4.3.2

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

$e^{\frac{t}{25}} \frac{d w}{d t} + \frac{e^{\frac{t}{25}} w}{25} = 10 e^{- \frac{t}{25} + \frac{t}{25}}$

Step 3.4.4

Add $- \frac{t}{25}$ and $\frac{t}{25}$ .

$e^{\frac{t}{25}} \frac{d w}{d t} + \frac{e^{\frac{t}{25}} w}{25} = 10 e^{0}$

Step 3.5

Simplify $10 e^{0}$ .

$e^{\frac{t}{25}} \frac{d w}{d t} + \frac{e^{\frac{t}{25}} w}{25} = 10$

Step 3.6

Reorder factors in $e^{\frac{t}{25}} \frac{d w}{d t} + \frac{e^{\frac{t}{25}} w}{25} = 10$ .

$e^{\frac{t}{25}} \frac{d w}{d t} + \frac{w e^{\frac{t}{25}}}{25} = 10$

Step 4

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

$\frac{d}{d t} [e^{\frac{t}{25}} w] = 10$

Step 5

Set up an integral on each side.

$\int \frac{d}{d t} [e^{\frac{t}{25}} w] d t = \int 10 d t$

Step 6

Integrate the left side.

$e^{\frac{t}{25}} w = \int 10 d t$

Step 7

Apply the constant rule.

$e^{\frac{t}{25}} w = 10 t + C$

Step 8

Divide each term in $e^{\frac{t}{25}} w = 10 t + C$ by $e^{\frac{t}{25}}$ and simplify.

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Step 8.1

Divide each term in $e^{\frac{t}{25}} w = 10 t + C$ by $e^{\frac{t}{25}}$ .

$\frac{e^{\frac{t}{25}} w}{e^{\frac{t}{25}}} = \frac{10 t}{e^{\frac{t}{25}}} + \frac{C}{e^{\frac{t}{25}}}$

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.

$\frac{e^{\frac{t}{25}} w}{e^{\frac{t}{25}}} = \frac{10 t}{e^{\frac{t}{25}}} + \frac{C}{e^{\frac{t}{25}}}$

Step 8.2.1.2

Divide $w$ by $1$ .

$w = \frac{10 t}{e^{\frac{t}{25}}} + \frac{C}{e^{\frac{t}{25}}}$