Calculus Examples

$\frac{d y}{d t} - y = 1$ , $y (0) = 1$

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.

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

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 $e^{- t}$ .

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

Multiply $e^{- t}$ by $1$ .

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

Step 2.4

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

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

Step 3

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

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

Step 4

Set up an integral on each side.

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

Step 5

Integrate the left side.

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

Step 6

Integrate the right side.

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Step 6.1

Let $u = - t$ . Then $d u = - d t$ , so $- d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1.1

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

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Step 6.1.1.1

Differentiate $- t$ .

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

Step 6.1.1.2

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

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

Step 6.1.1.3

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

$- 1 \cdot 1$

Step 6.1.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 6.1.2

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

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

Step 6.2

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

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

Step 6.3

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$e^{- t} y = - (e^{u} + C)$

Step 6.4

Simplify.

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

Step 6.5

Replace all occurrences of $u$ with $- t$ .

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

Step 7

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

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Step 7.1

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

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

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.

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

Step 7.2.1.2

Divide $y$ by $1$ .

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

Step 7.3

Simplify the right side.

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Step 7.3.1

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

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Step 7.3.1.1

Cancel the common factor.

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

Step 7.3.1.2

Divide $- 1$ by $1$ .

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

Step 8

Use the initial condition to find the value of $C$ by substituting $0$ for $t$ and $1$ for $y$ in $y = - 1 + \frac{C}{e^{- t}}$ .

$1 = - 1 + \frac{C}{e^{0}}$

Step 9

Solve for $C$ .

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Step 9.1

Rewrite the equation as $- 1 + \frac{C}{e^{0}} = 1$ .

$- 1 + \frac{C}{e^{0}} = 1$

Step 9.2

Simplify each term.

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Step 9.2.1

Anything raised to $0$ is $1$ .

$- 1 + \frac{C}{1} = 1$

Step 9.2.2

Divide $C$ by $1$ .

$- 1 + C = 1$

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 $1$ to both sides of the equation.

$C = 1 + 1$

Step 9.3.2

Add $1$ and $1$ .

$C = 2$

Step 10

Substitute $2$ for $C$ in $y = - 1 + \frac{C}{e^{- t}}$ and simplify.

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Step 10.1

Substitute $2$ for $C$ .

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