Calculus Examples

$\frac{d y}{d t} = e^{2 t} - 2 y$ , $y (1) = 2$

Step 1

Add $2 y$ to both sides of the equation.

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

Step 2

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

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Step 2.1

Set up the integration.

$e^{\int 2 d t}$

Step 2.2

Apply the constant rule.

$e^{2 t + C}$

Step 2.3

Remove the constant of integration.

$e^{2 t}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Multiply $e^{2 t}$ by $e^{2 t}$ by adding the exponents.

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Step 3.3.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.3.2

Add $2 t$ and $2 t$ .

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

Step 3.4

Reorder factors in $e^{2 t} \frac{d y}{d t} + 2 e^{2 t} y = e^{4 t}$ .

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Let $u = 4 t$ . Then $d u = 4 d t$ , so $\frac{1}{4} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.1.1

Let $u = 4 t$ . Find $\frac{d u}{d t}$ .

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Step 7.1.1.1

Differentiate $4 t$ .

$\frac{d}{d t} [4 t]$

Step 7.1.1.2

Since $4$ is constant with respect to $t$ , the derivative of $4 t$ with respect to $t$ is $4 \frac{d}{d t} [t]$ .

$4 \frac{d}{d t} [t]$

Step 7.1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$4 \cdot 1$

Step 7.1.1.4

Multiply $4$ by $1$ .

$4$

Step 7.1.2

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

$e^{2 t} y = \int e^{u} \frac{1}{4} d u$

Step 7.2

Combine $e^{u}$ and $\frac{1}{4}$ .

$e^{2 t} y = \int \frac{e^{u}}{4} d u$

Step 7.3

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$e^{2 t} y = \frac{1}{4} \int e^{u} d u$

Step 7.4

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

$e^{2 t} y = \frac{1}{4} (e^{u} + C)$

Step 7.5

Simplify.

$e^{2 t} y = \frac{1}{4} e^{u} + C$

Step 7.6

Replace all occurrences of $u$ with $4 t$ .

$e^{2 t} y = \frac{1}{4} e^{4 t} + C$

Step 8

Divide each term in $e^{2 t} y = \frac{1}{4} e^{4 t} + C$ by $e^{2 t}$ and simplify.

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Step 8.1

Divide each term in $e^{2 t} y = \frac{1}{4} e^{4 t} + C$ by $e^{2 t}$ .

$\frac{e^{2 t} y}{e^{2 t}} = \frac{\frac{1}{4} e^{4 t}}{e^{2 t}} + \frac{C}{e^{2 t}}$

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

$\frac{e^{2 t} y}{e^{2 t}} = \frac{\frac{1}{4} e^{4 t}}{e^{2 t}} + \frac{C}{e^{2 t}}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{4} e^{4 t}}{e^{2 t}} + \frac{C}{e^{2 t}}$

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 $e^{4 t}$ and $e^{2 t}$ .

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Step 8.3.1.1.1

Factor $e^{2 t}$ out of $\frac{1}{4} e^{4 t}$ .

$y = \frac{e^{2 t} (\frac{1}{4} e^{2 t})}{e^{2 t}} + \frac{C}{e^{2 t}}$

Step 8.3.1.1.2

Cancel the common factors.

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Step 8.3.1.1.2.1

Multiply by $1$ .

$y = \frac{e^{2 t} (\frac{1}{4} e^{2 t})}{e^{2 t} \cdot 1} + \frac{C}{e^{2 t}}$

Step 8.3.1.1.2.2

Cancel the common factor.

$y = \frac{e^{2 t} (\frac{1}{4} e^{2 t})}{e^{2 t} \cdot 1} + \frac{C}{e^{2 t}}$

Step 8.3.1.1.2.3

Rewrite the expression.

$y = \frac{\frac{1}{4} e^{2 t}}{1} + \frac{C}{e^{2 t}}$

Step 8.3.1.1.2.4

Divide $\frac{1}{4} e^{2 t}$ by $1$ .

$y = \frac{1}{4} e^{2 t} + \frac{C}{e^{2 t}}$

Step 8.3.1.2

Combine $\frac{1}{4}$ and $e^{2 t}$ .

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

Step 9

Use the initial condition to find the value of $C$ by substituting $1$ for $t$ and $2$ for $y$ in $y = \frac{e^{2 t}}{4} + \frac{C}{e^{2 t}}$ .

$2 = \frac{e^{2 \cdot 1}}{4} + \frac{C}{e^{2 \cdot 1}}$

Step 10

Solve for $C$ .

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Step 10.1

Rewrite the equation as $\frac{e^{2 \cdot 1}}{4} + \frac{C}{e^{2 \cdot 1}} = 2$ .

$\frac{e^{2 \cdot 1}}{4} + \frac{C}{e^{2 \cdot 1}} = 2$

Step 10.2

Simplify $\frac{e^{2 \cdot 1}}{4} + \frac{C}{e^{2 \cdot 1}}$ .

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Step 10.2.1

Multiply $2$ by $1$ .

$\frac{e^{2}}{4} + \frac{C}{e^{2 \cdot 1}} = 2$

Step 10.2.2

Multiply $2$ by $1$ .

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

Step 10.3

Subtract $\frac{e^{2}}{4}$ from both sides of the equation.

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

Step 10.4

Multiply both sides of the equation by $e^{2}$ .

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

Step 10.5

Simplify both sides of the equation.

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Step 10.5.1

Simplify the left side.

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Step 10.5.1.1

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

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Step 10.5.1.1.1

Cancel the common factor.

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

Step 10.5.1.1.2

Rewrite the expression.

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

Step 10.5.2

Simplify the right side.

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Step 10.5.2.1

Simplify $e^{2} (2 - \frac{e^{2}}{4})$ .

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Step 10.5.2.1.1

Apply the distributive property.

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

Step 10.5.2.1.2

Move $2$ to the left of $e^{2}$ .

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

Step 10.5.2.1.3

Multiply $e^{2} (- \frac{e^{2}}{4})$ .

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Step 10.5.2.1.3.1

Combine $e^{2}$ and $\frac{e^{2}}{4}$ .

$C = 2 \cdot e^{2} - \frac{e^{2} e^{2}}{4}$

Step 10.5.2.1.3.2

Multiply $e^{2}$ by $e^{2}$ by adding the exponents.

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Step 10.5.2.1.3.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 10.5.2.1.3.2.2

Add $2$ and $2$ .

$C = 2 \cdot e^{2} - \frac{e^{4}}{4}$

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

Step 11

Substitute $2 e^{2} - \frac{e^{4}}{4}$ for $C$ in $y = \frac{e^{2 t}}{4} + \frac{C}{e^{2 t}}$ and simplify.

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Step 11.1

Substitute $2 e^{2} - \frac{e^{4}}{4}$ for $C$ .

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

Step 11.2

Simplify each term.

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Step 11.2.1

Simplify the numerator.

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Step 11.2.1.1

To write $2 e^{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{e^{2 t}}{4} + \frac{2 e^{2} \cdot \frac{4}{4} - \frac{e^{4}}{4}}{e^{2 t}}$

Step 11.2.1.2

Combine $2 e^{2}$ and $\frac{4}{4}$ .

$y = \frac{e^{2 t}}{4} + \frac{\frac{2 e^{2} \cdot 4}{4} - \frac{e^{4}}{4}}{e^{2 t}}$

Step 11.2.1.3

Combine the numerators over the common denominator.

$y = \frac{e^{2 t}}{4} + \frac{\frac{2 e^{2} \cdot 4 - e^{4}}{4}}{e^{2 t}}$

Step 11.2.1.4

Multiply $4$ by $2$ .

$y = \frac{e^{2 t}}{4} + \frac{\frac{8 e^{2} - e^{4}}{4}}{e^{2 t}}$

Step 11.2.2

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{e^{2 t}}{4} + \frac{8 e^{2} - e^{4}}{4} \cdot \frac{1}{e^{2 t}}$

Step 11.2.3

Multiply $\frac{8 e^{2} - e^{4}}{4}$ by $\frac{1}{e^{2 t}}$ .

$y = \frac{e^{2 t}}{4} + \frac{8 e^{2} - e^{4}}{4 e^{2 t}}$

Step 11.3

To write $\frac{e^{2 t}}{4}$ as a fraction with a common denominator, multiply by $\frac{e^{2 t}}{e^{2 t}}$ .

$y = \frac{e^{2 t}}{4} \cdot \frac{e^{2 t}}{e^{2 t}} + \frac{8 e^{2} - e^{4}}{4 e^{2 t}}$

Step 11.4

Multiply $\frac{e^{2 t}}{4}$ by $\frac{e^{2 t}}{e^{2 t}}$ .

$y = \frac{e^{2 t} e^{2 t}}{4 e^{2 t}} + \frac{8 e^{2} - e^{4}}{4 e^{2 t}}$

Step 11.5

Combine the numerators over the common denominator.

$y = \frac{e^{2 t} e^{2 t} + 8 e^{2} - e^{4}}{4 e^{2 t}}$

Step 11.6

Multiply $e^{2 t}$ by $e^{2 t}$ by adding the exponents.

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Step 11.6.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{e^{2 t + 2 t} + 8 e^{2} - e^{4}}{4 e^{2 t}}$

Step 11.6.2

Add $2 t$ and $2 t$ .

$y = \frac{e^{4 t} + 8 e^{2} - e^{4}}{4 e^{2 t}}$