Calculus Examples

$t \frac{d x}{d t} + 2 x = 4 e^{t}$

Step 1

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

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Step 1.1

Divide each term in $t \frac{d x}{d t} + 2 x = 4 e^{t}$ by $t$ .

$\frac{t \frac{d x}{d t}}{t} + \frac{2 x}{t} = \frac{4 e^{t}}{t}$

Step 1.2

Cancel the common factor of $t$ .

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Step 1.2.1

Cancel the common factor.

$\frac{t \frac{d x}{d t}}{t} + \frac{2 x}{t} = \frac{4 e^{t}}{t}$

Step 1.2.2

Divide $\frac{d x}{d t}$ by $1$ .

$\frac{d x}{d t} + \frac{2 x}{t} = \frac{4 e^{t}}{t}$

Step 1.3

Factor $x$ out of $\frac{2 x}{t}$ .

$\frac{d x}{d t} + x \frac{2}{t} = \frac{4 e^{t}}{t}$

Step 1.4

Reorder $x$ and $\frac{2}{t}$ .

$\frac{d x}{d t} + \frac{2}{t} x = \frac{4 e^{t}}{t}$

Step 2

The integrating factor is defined by the formula $e^{\int P (t) d t}$ , where $P (t) = \frac{2}{t}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{2}{t} d t}$

Step 2.2

Integrate $\frac{2}{t}$ .

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Step 2.2.1

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

$e^{2 \int \frac{1}{t} d t}$

Step 2.2.2

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

$e^{2 (\ln (| t |) + C)}$

Step 2.2.3

Simplify.

$e^{2 \ln (| t |) + C}$

Step 2.3

Remove the constant of integration.

$e^{2 \ln (t)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (t^{2})}$

Step 2.5

Exponentiation and log are inverse functions.

$t^{2}$

Step 3

Multiply each term by the integrating factor $t^{2}$ .

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Step 3.1

Multiply each term by $t^{2}$ .

$t^{2} \frac{d x}{d t} + t^{2} (\frac{2}{t} x) = t^{2} \frac{4 e^{t}}{t}$

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $\frac{2}{t}$ and $x$ .

$t^{2} \frac{d x}{d t} + t^{2} \frac{2 x}{t} = t^{2} \frac{4 e^{t}}{t}$

Step 3.2.2

Cancel the common factor of $t$ .

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Step 3.2.2.1

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

$t^{2} \frac{d x}{d t} + t \cdot t \frac{2 x}{t} = t^{2} \frac{4 e^{t}}{t}$

Step 3.2.2.2

Cancel the common factor.

$t^{2} \frac{d x}{d t} + t \cdot t \frac{2 x}{t} = t^{2} \frac{4 e^{t}}{t}$

Step 3.2.2.3

Rewrite the expression.

$t^{2} \frac{d x}{d t} + t (2 x) = t^{2} \frac{4 e^{t}}{t}$

Step 3.2.3

Rewrite using the commutative property of multiplication.

$t^{2} \frac{d x}{d t} + 2 t x = t^{2} \frac{4 e^{t}}{t}$

Step 3.3

Cancel the common factor of $t$ .

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Step 3.3.1

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

$t^{2} \frac{d x}{d t} + 2 t x = t \cdot t \frac{4 e^{t}}{t}$

Step 3.3.2

Cancel the common factor.

$t^{2} \frac{d x}{d t} + 2 t x = t \cdot t \frac{4 e^{t}}{t}$

Step 3.3.3

Rewrite the expression.

$t^{2} \frac{d x}{d t} + 2 t x = t (4 e^{t})$

Step 3.4

Rewrite using the commutative property of multiplication.

$t^{2} \frac{d x}{d t} + 2 t x = 4 t e^{t}$

Step 4

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

$\frac{d}{d t} [t^{2} x] = 4 t e^{t}$

Step 5

Set up an integral on each side.

$\int \frac{d}{d t} [t^{2} x] d t = \int 4 t e^{t} d t$

Step 6

Integrate the left side.

$t^{2} x = \int 4 t e^{t} d t$

Step 7

Integrate the right side.

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Step 7.1

Since $4$ is constant with respect to $t$ , move $4$ out of the integral.

$t^{2} x = 4 \int t e^{t} d t$

Step 7.2

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

$t^{2} x = 4 (t e^{t} - \int e^{t} d t)$

Step 7.3

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

$t^{2} x = 4 (t e^{t} - (e^{t} + C))$

Step 7.4

Simplify.

$t^{2} x = 4 (t e^{t} - e^{t}) + C$

Step 8

Divide each term in $t^{2} x = 4 (t e^{t} - e^{t}) + C$ by $t^{2}$ and simplify.

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Step 8.1

Divide each term in $t^{2} x = 4 (t e^{t} - e^{t}) + C$ by $t^{2}$ .

$\frac{t^{2} x}{t^{2}} = \frac{4 (t e^{t} - e^{t})}{t^{2}} + \frac{C}{t^{2}}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $t^{2}$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{t^{2} x}{t^{2}} = \frac{4 (t e^{t} - e^{t})}{t^{2}} + \frac{C}{t^{2}}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{4 (t e^{t} - e^{t})}{t^{2}} + \frac{C}{t^{2}}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Combine the numerators over the common denominator.

$x = \frac{4 (t e^{t} - e^{t}) + C}{t^{2}}$

Step 8.3.2

Simplify the numerator.

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Step 8.3.2.1

Apply the distributive property.

$x = \frac{4 (t e^{t}) + 4 (- e^{t}) + C}{t^{2}}$

Step 8.3.2.2

Multiply $- 1$ by $4$ .

$x = \frac{4 t e^{t} - 4 e^{t} + C}{t^{2}}$