Calculus Examples

$\frac{d y}{d t} = y + 5 t$

Step 1

Subtract $y$ from both sides of the equation.

$\frac{d y}{d t} - y = 5 t$

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 $e^{- t}$ .

$e^{- t} \frac{d y}{d t} + e^{- t} (- y) = e^{- t} (5 t)$

Step 3.2

Rewrite using the commutative property of multiplication.

$e^{- t} \frac{d y}{d t} - e^{- t} y = e^{- t} (5 t)$

Step 3.3

Rewrite using the commutative property of multiplication.

$e^{- t} \frac{d y}{d t} - e^{- t} y = 5 e^{- t} t$

Step 3.4

Reorder factors in $e^{- t} \frac{d y}{d t} - e^{- t} y = 5 e^{- t} t$ .

$e^{- t} \frac{d y}{d t} - y e^{- t} = 5 t e^{- t}$

Step 4

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

$\frac{d}{d t} [e^{- t} y] = 5 t e^{- t}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d t} [e^{- t} y] d t = \int 5 t e^{- t} d t$

Step 6

Integrate the left side.

$e^{- t} y = \int 5 t e^{- t} d t$

Step 7

Integrate the right side.

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Step 7.1

Since $5$ is constant with respect to $t$ , move $5$ out of the integral.

$e^{- t} y = 5 \int t e^{- t} d t$

Step 7.2

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

$e^{- t} y = 5 (t (- e^{- t}) - \int - e^{- t} d t)$

Step 7.3

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

$e^{- t} y = 5 (t (- e^{- t}) - - \int e^{- t} d t)$

Step 7.4

Simplify.

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Step 7.4.1

Multiply $- 1$ by $- 1$ .

$e^{- t} y = 5 (t (- e^{- t}) + 1 \int e^{- t} d t)$

Step 7.4.2

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

$e^{- t} y = 5 (t (- e^{- t}) + \int e^{- t} d t)$

Step 7.5

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

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Step 7.5.1

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

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Step 7.5.1.1

Differentiate $- t$ .

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

Step 7.5.1.2

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 7.5.1.3

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 7.5.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 7.5.2

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

$e^{- t} y = 5 (t (- e^{- t}) + \int - e^{u} d u)$

Step 7.6

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

$e^{- t} y = 5 (t (- e^{- t}) - \int e^{u} d u)$

Step 7.7

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

$e^{- t} y = 5 (t (- e^{- t}) - (e^{u} + C))$

Step 7.8

Rewrite $5 (t (- e^{- t}) - (e^{u} + C))$ as $5 (- t e^{- t} - e^{u}) + C$ .

$e^{- t} y = 5 (- t e^{- t} - e^{u}) + C$

Step 7.9

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

$e^{- t} y = 5 (- t e^{- t} - e^{- t}) + C$

Step 8

Divide each term in $e^{- t} y = 5 (- t e^{- t} - e^{- t}) + C$ by $e^{- t}$ and simplify.

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Step 8.1

Divide each term in $e^{- t} y = 5 (- t e^{- t} - e^{- t}) + C$ by $e^{- t}$ .

$\frac{e^{- t} y}{e^{- t}} = \frac{5 (- t e^{- t} - e^{- t})}{e^{- t}} + \frac{C}{e^{- t}}$

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.

$\frac{e^{- t} y}{e^{- t}} = \frac{5 (- t e^{- t} - e^{- t})}{e^{- t}} + \frac{C}{e^{- t}}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{5 (- t e^{- t} - e^{- t})}{e^{- t}} + \frac{C}{e^{- t}}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Combine the numerators over the common denominator.

$y = \frac{5 (- t e^{- t} - e^{- t}) + C}{e^{- t}}$

Step 8.3.2

Simplify the numerator.

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Step 8.3.2.1

Apply the distributive property.

$y = \frac{5 (- t e^{- t}) + 5 (- e^{- t}) + C}{e^{- t}}$

Step 8.3.2.2

Multiply $- 1$ by $5$ .

$y = \frac{- 5 (t e^{- t}) + 5 (- e^{- t}) + C}{e^{- t}}$

Step 8.3.2.3

Multiply $- 1$ by $5$ .

$y = \frac{- 5 t e^{- t} - 5 e^{- t} + C}{e^{- t}}$

Step 8.3.3

Simplify with factoring out.

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Step 8.3.3.1

Factor $- 1$ out of $- 5 t e^{- t}$ .

$y = \frac{- (5 t e^{- t}) - 5 e^{- t} + C}{e^{- t}}$

Step 8.3.3.2

Factor $- 1$ out of $- 5 e^{- t}$ .

$y = \frac{- (5 t e^{- t}) - (5 e^{- t}) + C}{e^{- t}}$

Step 8.3.3.3

Factor $- 1$ out of $- (5 t e^{- t}) - (5 e^{- t})$ .

$y = \frac{- (5 t e^{- t} + 5 e^{- t}) + C}{e^{- t}}$

Step 8.3.3.4

Factor $- 1$ out of $C$ .

$y = \frac{- (5 t e^{- t} + 5 e^{- t}) - 1 (- C)}{e^{- t}}$

Step 8.3.3.5

Factor $- 1$ out of $- (5 t e^{- t} + 5 e^{- t}) - 1 (- C)$ .

$y = \frac{- (5 t e^{- t} + 5 e^{- t} - C)}{e^{- t}}$

Step 8.3.3.6

Simplify the expression.

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Step 8.3.3.6.1

Rewrite $- (5 t e^{- t} + 5 e^{- t} - C)$ as $- 1 (5 t e^{- t} + 5 e^{- t} - C)$ .

$y = \frac{- 1 (5 t e^{- t} + 5 e^{- t} - C)}{e^{- t}}$

Step 8.3.3.6.2

Move the negative in front of the fraction.

$y = - \frac{5 t e^{- t} + 5 e^{- t} - C}{e^{- t}}$