Calculus Examples

$\frac{d y}{d t} + y = e^{- t}$

Step 1

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

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Step 1.1

Set up the integration.

$e^{\int d t}$

Step 1.2

Apply the constant rule.

$e^{t + C}$

Step 1.3

Remove the constant of integration.

$e^{t}$

Step 2

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

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Step 2.1

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

$e^{t} \frac{d y}{d t} + e^{t} y = e^{t} e^{- t}$

Step 2.2

Multiply $e^{t}$ by $e^{- t}$ by adding the exponents.

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Step 2.2.1

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

$e^{t} \frac{d y}{d t} + e^{t} y = e^{t - t}$

Step 2.2.2

Subtract $t$ from $t$ .

$e^{t} \frac{d y}{d t} + e^{t} y = e^{0}$

Step 2.3

Simplify $e^{0}$ .

$e^{t} \frac{d y}{d t} + e^{t} y = 1$

Step 2.4

Reorder factors in $e^{t} \frac{d y}{d t} + e^{t} y = 1$ .

$e^{t} \frac{d y}{d t} + y e^{t} = 1$

Step 3

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

$\frac{d}{d t} [e^{t} y] = 1$

Step 4

Set up an integral on each side.

$\int \frac{d}{d t} [e^{t} y] d t = \int d t$

Step 5

Integrate the left side.

$e^{t} y = \int d t$

Step 6

Apply the constant rule.

$e^{t} y = t + C$

Step 7

Divide each term in $e^{t} y = t + C$ by $e^{t}$ and simplify.

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Step 7.1

Divide each term in $e^{t} y = t + C$ by $e^{t}$ .

$\frac{e^{t} y}{e^{t}} = \frac{t}{e^{t}} + \frac{C}{e^{t}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

$\frac{e^{t} y}{e^{t}} = \frac{t}{e^{t}} + \frac{C}{e^{t}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{t}{e^{t}} + \frac{C}{e^{t}}$