Calculus Examples

$\frac{d y}{d t} = t^{2} - y$

Step 1

Add $y$ to both sides of the equation.

$\frac{d y}{d t} + y = t^{2}$

Step 2

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

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Step 2.1

Set up the integration.

$e^{\int d t}$

Step 2.2

Apply the constant rule.

$e^{t + C}$

Step 2.3

Remove the constant of integration.

$e^{t}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Reorder factors in $e^{t} \frac{d y}{d t} + e^{t} y = e^{t} t^{2}$ .

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

Step 4

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

$\frac{d}{d t} [e^{t} y] = t^{2} e^{t}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d t} [e^{t} y] d t = \int t^{2} e^{t} d t$

Step 6

Integrate the left side.

$e^{t} y = \int t^{2} e^{t} d t$

Step 7

Integrate the right side.

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Step 7.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = t^{2}$ and $d v = e^{t}$ .

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

Step 7.2

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

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

Step 7.3

Multiply $2$ by $- 1$ .

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

Step 7.4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = t$ and $d v = e^{t}$ .

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

Step 7.5

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

$e^{t} y = t^{2} e^{t} - 2 (t e^{t} - (e^{t} + C))$

Step 7.6

Rewrite $t^{2} e^{t} - 2 (t e^{t} - (e^{t} + C))$ as $t^{2} e^{t} - 2 t e^{t} + 2 e^{t} + C$ .

$e^{t} y = t^{2} e^{t} - 2 t e^{t} + 2 e^{t} + C$

Step 8

Divide each term in $e^{t} y = t^{2} e^{t} - 2 t e^{t} + 2 e^{t} + C$ by $e^{t}$ and simplify.

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Step 8.1

Divide each term in $e^{t} y = t^{2} e^{t} - 2 t e^{t} + 2 e^{t} + C$ by $e^{t}$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

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

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

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Step 8.3.1.1.1

Cancel the common factor.

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

Step 8.3.1.1.2

Divide $t^{2}$ by $1$ .

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

Step 8.3.1.2

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

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Step 8.3.1.2.1

Cancel the common factor.

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

Step 8.3.1.2.2

Divide $- 2 t$ by $1$ .

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

Step 8.3.1.3

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

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Step 8.3.1.3.1

Cancel the common factor.

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

Step 8.3.1.3.2

Divide $2$ by $1$ .

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