Calculus Examples

$\frac{d y}{d t} = y + t^{3}$

Step 1

Subtract $y$ from both sides of the equation.

$\frac{d y}{d t} - y = t^{3}$

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.

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

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 $e^{- t}$ .

$e^{- t} \frac{d y}{d t} + e^{- t} (- y) = e^{- t} t^{3}$

Step 3.2

Rewrite using the commutative property of multiplication.

$e^{- t} \frac{d y}{d t} - e^{- t} y = e^{- t} t^{3}$

Step 3.3

Reorder factors in $e^{- t} \frac{d y}{d t} - e^{- t} y = e^{- t} t^{3}$ .

$e^{- t} \frac{d y}{d t} - y e^{- t} = t^{3} e^{- t}$

Step 4

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

$\frac{d}{d t} [e^{- t} y] = t^{3} e^{- t}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d t} [e^{- t} y] d t = \int t^{3} e^{- t} d t$

Step 6

Integrate the left side.

$e^{- t} y = \int t^{3} e^{- t} d t$

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 $u = t^{3}$ and $d v = e^{- t}$ .

$e^{- t} y = t^{3} (- e^{- t}) - \int - e^{- t} (3 t^{2}) d t$

Step 7.2

Multiply $3$ by $- 1$ .

$e^{- t} y = t^{3} (- e^{- t}) - \int - 3 e^{- t} t^{2} d t$

Step 7.3

Since $- 3$ is constant with respect to $t$ , move $- 3$ out of the integral.

$e^{- t} y = t^{3} (- e^{- t}) - (- 3 \int e^{- t} t^{2} d t)$

Step 7.4

Multiply $- 3$ by $- 1$ .

$e^{- t} y = t^{3} (- e^{- t}) + 3 \int e^{- t} t^{2} d t$

Step 7.5

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

$e^{- t} y = t^{3} (- e^{- t}) + 3 (t^{2} (- e^{- t}) - \int - e^{- t} (2 t) d t)$

Step 7.6

Multiply $2$ by $- 1$ .

$e^{- t} y = t^{3} (- e^{- t}) + 3 (t^{2} (- e^{- t}) - \int - 2 e^{- t} t d t)$

Step 7.7

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

$e^{- t} y = t^{3} (- e^{- t}) + 3 (t^{2} (- e^{- t}) - (- 2 \int e^{- t} t d t))$

Step 7.8

Multiply $- 2$ by $- 1$ .

$e^{- t} y = t^{3} (- e^{- t}) + 3 (t^{2} (- e^{- t}) + 2 \int e^{- t} t d t)$

Step 7.9

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

$e^{- t} y = t^{3} (- e^{- t}) + 3 (t^{2} (- e^{- t}) + 2 (t (- e^{- t}) - \int - e^{- t} d t))$

Step 7.10

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

$e^{- t} y = t^{3} (- e^{- t}) + 3 (t^{2} (- e^{- t}) + 2 (t (- e^{- t}) - - \int e^{- t} d t))$

Step 7.11

Simplify.

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Step 7.11.1

Multiply $- 1$ by $- 1$ .

$e^{- t} y = t^{3} (- e^{- t}) + 3 (t^{2} (- e^{- t}) + 2 (t (- e^{- t}) + 1 \int e^{- t} d t))$

Step 7.11.2

Multiply $\int e^{- t} d t$ by $1$ .

$e^{- t} y = t^{3} (- e^{- t}) + 3 (t^{2} (- e^{- t}) + 2 (t (- e^{- t}) + \int e^{- t} d t))$

Step 7.12

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

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Step 7.12.1

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

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Step 7.12.1.1

Differentiate $- t$ .

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

Step 7.12.1.2

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 7.12.1.3

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 7.12.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 7.12.2

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

$e^{- t} y = t^{3} (- e^{- t}) + 3 (t^{2} (- e^{- t}) + 2 (t (- e^{- t}) + \int - e^{u} d u))$

Step 7.13

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

$e^{- t} y = t^{3} (- e^{- t}) + 3 (t^{2} (- e^{- t}) + 2 (t (- e^{- t}) - \int e^{u} d u))$

Step 7.14

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

$e^{- t} y = t^{3} (- e^{- t}) + 3 (t^{2} (- e^{- t}) + 2 (t (- e^{- t}) - (e^{u} + C)))$

Step 7.15

Rewrite $t^{3} (- e^{- t}) + 3 (t^{2} (- e^{- t}) + 2 (t (- e^{- t}) - (e^{u} + C)))$ as $- t^{3} e^{- t} + 3 (t^{2} (- e^{- t}) + 2 (t (- e^{- t}) - e^{u})) + C$ .

$e^{- t} y = - t^{3} e^{- t} + 3 (t^{2} (- e^{- t}) + 2 (t (- e^{- t}) - e^{u})) + C$

Step 7.16

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

$e^{- t} y = - t^{3} e^{- t} + 3 (t^{2} (- e^{- t}) + 2 (t (- e^{- t}) - e^{- t})) + C$

Step 8

Divide each term in $e^{- t} y = - t^{3} e^{- t} + 3 (t^{2} (- e^{- t}) + 2 (t (- e^{- t}) - e^{- t})) + C$ by $e^{- t}$ and simplify.

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Step 8.1

Divide each term in $e^{- t} y = - t^{3} e^{- t} + 3 (t^{2} (- e^{- t}) + 2 (t (- e^{- t}) - e^{- t})) + C$ by $e^{- t}$ .

$\frac{e^{- t} y}{e^{- t}} = \frac{- t^{3} e^{- t}}{e^{- t}} + \frac{3 (t^{2} (- e^{- t}) + 2 (t (- e^{- t}) - e^{- t}))}{e^{- t}} + \frac{C}{e^{- t}}$

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.

$\frac{e^{- t} y}{e^{- t}} = \frac{- t^{3} e^{- t}}{e^{- t}} + \frac{3 (t^{2} (- e^{- t}) + 2 (t (- e^{- t}) - e^{- t}))}{e^{- t}} + \frac{C}{e^{- t}}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{- t^{3} e^{- t}}{e^{- t}} + \frac{3 (t^{2} (- e^{- t}) + 2 (t (- e^{- t}) - e^{- t}))}{e^{- t}} + \frac{C}{e^{- t}}$

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.

$y = \frac{- t^{3} e^{- t}}{e^{- t}} + \frac{3 (t^{2} (- e^{- t}) + 2 (t (- e^{- t}) - e^{- t}))}{e^{- t}} + \frac{C}{e^{- t}}$

Step 8.3.1.1.2

Divide $- t^{3}$ by $1$ .

$y = - t^{3} + \frac{3 (t^{2} (- e^{- t}) + 2 (t (- e^{- t}) - e^{- t}))}{e^{- t}} + \frac{C}{e^{- t}}$

Step 8.3.1.2

Simplify the numerator.

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Step 8.3.1.2.1

Rewrite using the commutative property of multiplication.

$y = - t^{3} + \frac{3 (- t^{2} e^{- t} + 2 (t (- e^{- t}) - e^{- t}))}{e^{- t}} + \frac{C}{e^{- t}}$

Step 8.3.1.2.2

Rewrite using the commutative property of multiplication.

$y = - t^{3} + \frac{3 (- t^{2} e^{- t} + 2 (- t e^{- t} - e^{- t}))}{e^{- t}} + \frac{C}{e^{- t}}$

Step 8.3.1.2.3

Apply the distributive property.

$y = - t^{3} + \frac{3 (- t^{2} e^{- t} + 2 (- t e^{- t}) + 2 (- e^{- t}))}{e^{- t}} + \frac{C}{e^{- t}}$

Step 8.3.1.2.4

Multiply $- 1$ by $2$ .

$y = - t^{3} + \frac{3 (- t^{2} e^{- t} - 2 (t e^{- t}) + 2 (- e^{- t}))}{e^{- t}} + \frac{C}{e^{- t}}$

Step 8.3.1.2.5

Multiply $- 1$ by $2$ .

$y = - t^{3} + \frac{3 (- t^{2} e^{- t} - 2 (t e^{- t}) - 2 e^{- t})}{e^{- t}} + \frac{C}{e^{- t}}$

Step 8.3.1.2.6

Factor $e^{- t}$ out of $- t^{2} e^{- t} - 2 t e^{- t} - 2 e^{- t}$ .

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Step 8.3.1.2.6.1

Factor $e^{- t}$ out of $- t^{2} e^{- t}$ .

$y = - t^{3} + \frac{3 (e^{- t} (- t^{2}) - 2 t e^{- t} - 2 e^{- t})}{e^{- t}} + \frac{C}{e^{- t}}$

Step 8.3.1.2.6.2

Factor $e^{- t}$ out of $- 2 t e^{- t}$ .

$y = - t^{3} + \frac{3 (e^{- t} (- t^{2}) + e^{- t} (- 2 t) - 2 e^{- t})}{e^{- t}} + \frac{C}{e^{- t}}$

Step 8.3.1.2.6.3

Factor $e^{- t}$ out of $- 2 e^{- t}$ .

$y = - t^{3} + \frac{3 (e^{- t} (- t^{2}) + e^{- t} (- 2 t) + e^{- t} \cdot - 2)}{e^{- t}} + \frac{C}{e^{- t}}$

Step 8.3.1.2.6.4

Factor $e^{- t}$ out of $e^{- t} (- t^{2}) + e^{- t} (- 2 t)$ .

$y = - t^{3} + \frac{3 (e^{- t} (- t^{2} - 2 t) + e^{- t} \cdot - 2)}{e^{- t}} + \frac{C}{e^{- t}}$

Step 8.3.1.2.6.5

Factor $e^{- t}$ out of $e^{- t} (- t^{2} - 2 t) + e^{- t} \cdot - 2$ .

$y = - t^{3} + \frac{3 (e^{- t} (- t^{2} - 2 t - 2))}{e^{- t}} + \frac{C}{e^{- t}}$

$y = - t^{3} + \frac{3 e^{- t} (- t^{2} - 2 t - 2)}{e^{- t}} + \frac{C}{e^{- t}}$

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.

$y = - t^{3} + \frac{3 e^{- t} (- t^{2} - 2 t - 2)}{e^{- t}} + \frac{C}{e^{- t}}$

Step 8.3.1.3.2

Divide $3 (- t^{2} - 2 t - 2)$ by $1$ .

$y = - t^{3} + 3 (- t^{2} - 2 t - 2) + \frac{C}{e^{- t}}$

Step 8.3.1.4

Apply the distributive property.

$y = - t^{3} + 3 (- t^{2}) + 3 (- 2 t) + 3 \cdot - 2 + \frac{C}{e^{- t}}$

Step 8.3.1.5

Simplify.

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Step 8.3.1.5.1

Multiply $- 1$ by $3$ .

$y = - t^{3} - 3 t^{2} + 3 (- 2 t) + 3 \cdot - 2 + \frac{C}{e^{- t}}$

Step 8.3.1.5.2

Multiply $- 2$ by $3$ .

$y = - t^{3} - 3 t^{2} - 6 t + 3 \cdot - 2 + \frac{C}{e^{- t}}$

Step 8.3.1.5.3

Multiply $3$ by $- 2$ .

$y = - t^{3} - 3 t^{2} - 6 t - 6 + \frac{C}{e^{- t}}$