Calculus Examples

$\frac{d s}{d t} + \frac{s}{t} = \cos (t) + \frac{\sin (t)}{t}$

Step 1

Rewrite the differential equation as $\frac{d s}{d t} + M (t) s = Q (t)$ .

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Step 1.1

Reorder terms.

$\frac{d s}{d t} + \frac{s}{t} = \frac{\sin (t)}{t} + \cos (t)$

Step 1.2

Factor $s$ out of $\frac{s}{t}$ .

$\frac{d s}{d t} + s \frac{1}{t} = \frac{\sin (t)}{t} + \cos (t)$

Step 1.3

Reorder $s$ and $\frac{1}{t}$ .

$\frac{d s}{d t} + \frac{1}{t} s = \frac{\sin (t)}{t} + \cos (t)$

Step 2

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

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Step 2.1

Set up the integration.

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

Step 2.2

The integral of $\frac{1}{t}$ with respect to $t$ is $\ln (| t |)$ .

$e^{\ln (| t |) + C}$

Step 2.3

Remove the constant of integration.

$e^{\ln (t)}$

Step 2.4

Exponentiation and log are inverse functions.

$t$

Step 3

Multiply each term by the integrating factor $t$ .

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Step 3.1

Multiply each term by $t$ .

$t \frac{d s}{d t} + t (\frac{1}{t} s) = t \frac{\sin (t)}{t} + t \cos (t)$

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $\frac{1}{t}$ and $s$ .

$t \frac{d s}{d t} + t \frac{s}{t} = t \frac{\sin (t)}{t} + t \cos (t)$

Step 3.2.2

Cancel the common factor of $t$ .

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Step 3.2.2.1

Cancel the common factor.

$t \frac{d s}{d t} + t \frac{s}{t} = t \frac{\sin (t)}{t} + t \cos (t)$

Step 3.2.2.2

Rewrite the expression.

$t \frac{d s}{d t} + s = t \frac{\sin (t)}{t} + t \cos (t)$

Step 3.3

Cancel the common factor of $t$ .

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Step 3.3.1

Cancel the common factor.

$t \frac{d s}{d t} + s = t \frac{\sin (t)}{t} + t \cos (t)$

Step 3.3.2

Rewrite the expression.

$t \frac{d s}{d t} + s = \sin (t) + t \cos (t)$

Step 4

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

$\frac{d}{d t} [t s] = \sin (t) + t \cos (t)$

Step 5

Set up an integral on each side.

$\int \frac{d}{d t} [t s] d t = \int \sin (t) + t \cos (t) d t$

Step 6

Integrate the left side.

$t s = \int \sin (t) + t \cos (t) d t$

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

$t s = \int \sin (t) d t + \int t \cos (t) d t$

Step 7.2

The integral of $\sin (t)$ with respect to $t$ is $- \cos (t)$ .

$t s = - \cos (t) + C + \int t \cos (t) d t$

Step 7.3

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = t$ and $d v = \cos (t)$ .

$t s = - \cos (t) + C + t \sin (t) - \int \sin (t) d t$

Step 7.4

The integral of $\sin (t)$ with respect to $t$ is $- \cos (t)$ .

$t s = - \cos (t) + C + t \sin (t) - (- \cos (t) + C)$

Step 7.5

Simplify.

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Step 7.5.1

Simplify.

$t s = - \cos (t) + t \sin (t) + \cos (t) + C$

Step 7.5.2

Simplify.

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Step 7.5.2.1

Add $- \cos (t)$ and $\cos (t)$ .

$t s = t \sin (t) + 0 + C$

Step 7.5.2.2

Add $t \sin (t)$ and $0$ .

$t s = t \sin (t) + C$

Step 8

Divide each term in $t s = t \sin (t) + C$ by $t$ and simplify.

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Step 8.1

Divide each term in $t s = t \sin (t) + C$ by $t$ .

$\frac{t s}{t} = \frac{t \sin (t)}{t} + \frac{C}{t}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $t$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{t s}{t} = \frac{t \sin (t)}{t} + \frac{C}{t}$

Step 8.2.1.2

Divide $s$ by $1$ .

$s = \frac{t \sin (t)}{t} + \frac{C}{t}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Cancel the common factor of $t$ .

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Step 8.3.1.1

Cancel the common factor.

$s = \frac{t \sin (t)}{t} + \frac{C}{t}$

Step 8.3.1.2

Divide $\sin (t)$ by $1$ .

$s = \sin (t) + \frac{C}{t}$