Calculus Examples

$y'''' + 2 y = 3 e^{t}$ with $y (0) = 3$

Step 1

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

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Step 1.1

Set up the integration.

$e^{\int 2 d t}$

Step 1.2

Apply the constant rule.

$e^{2 t + C}$

Step 1.3

Remove the constant of integration.

$e^{2 t}$

Step 2

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

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Step 2.1

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

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

Rewrite using the commutative property of multiplication.

$e^{2 t} \frac{d y}{d t} + 2 e^{2 t} y = 3 e^{2 t} e^{t}$

Step 2.4

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

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Step 2.4.1

Move $e^{t}$ .

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

Step 2.4.2

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

$e^{2 t} \frac{d y}{d t} + 2 e^{2 t} y = 3 e^{t + 2 t}$

Step 2.4.3

Add $t$ and $2 t$ .

$e^{2 t} \frac{d y}{d t} + 2 e^{2 t} y = 3 e^{3 t}$

Step 2.5

Reorder factors in $e^{2 t} \frac{d y}{d t} + 2 e^{2 t} y = 3 e^{3 t}$ .

$e^{2 t} \frac{d y}{d t} + 2 y e^{2 t} = 3 e^{3 t}$

Step 3

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

$\frac{d}{d t} [e^{2 t} y] = 3 e^{3 t}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d t} [e^{2 t} y] d t = \int 3 e^{3 t} d t$

Step 5

Integrate the left side.

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

Step 6

Integrate the right side.

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Step 6.1

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

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

Step 6.2

Let $u = 3 t$ . Then $d u = 3 d t$ , so $\frac{1}{3} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.2.1

Let $u = 3 t$ . Find $\frac{d u}{d t}$ .

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Step 6.2.1.1

Differentiate $3 t$ .

$\frac{d}{d t} [3 t]$

Step 6.2.1.2

Since $3$ is constant with respect to $t$ , the derivative of $3 t$ with respect to $t$ is $3 \frac{d}{d t} [t]$ .

$3 \frac{d}{d t} [t]$

Step 6.2.1.3

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

$3 \cdot 1$

Step 6.2.1.4

Multiply $3$ by $1$ .

$3$

Step 6.2.2

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

$e^{2 t} y = 3 \int e^{u} \frac{1}{3} d u$

Step 6.3

Combine $e^{u}$ and $\frac{1}{3}$ .

$e^{2 t} y = 3 \int \frac{e^{u}}{3} d u$

Step 6.4

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

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

Step 6.5

Simplify.

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Step 6.5.1

Combine $\frac{1}{3}$ and $3$ .

$e^{2 t} y = \frac{3}{3} \int e^{u} d u$

Step 6.5.2

Cancel the common factor of $3$ .

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Step 6.5.2.1

Cancel the common factor.

$e^{2 t} y = \frac{3}{3} \int e^{u} d u$

Step 6.5.2.2

Rewrite the expression.

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

Step 6.5.3

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

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

Step 6.6

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

$e^{2 t} y = e^{u} + C$

Step 6.7

Replace all occurrences of $u$ with $3 t$ .

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $y$ by $1$ .

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

Step 7.3

Simplify the right side.

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Step 7.3.1

Cancel the common factor of $e^{3 t}$ and $e^{2 t}$ .

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Step 7.3.1.1

Factor $e^{2 t}$ out of $e^{3 t}$ .

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

Step 7.3.1.2

Cancel the common factors.

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Step 7.3.1.2.1

Multiply by $1$ .

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

Step 7.3.1.2.2

Cancel the common factor.

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

Step 7.3.1.2.3

Rewrite the expression.

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

Step 7.3.1.2.4

Divide $e^{t}$ by $1$ .

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

Step 8

Use the initial condition to find the value of $C$ by substituting $0$ for $t$ and $3$ for $y$ in $y = e^{t} + \frac{C}{e^{2 t}}$ .

$3 = e^{0} + \frac{C}{e^{2 \cdot 0}}$

Step 9

Solve for $C$ .

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Step 9.1

Rewrite the equation as $e^{0} + \frac{C}{e^{2 \cdot 0}} = 3$ .

$e^{0} + \frac{C}{e^{2 \cdot 0}} = 3$

Step 9.2

Simplify each term.

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Step 9.2.1

Anything raised to $0$ is $1$ .

$1 + \frac{C}{e^{2 \cdot 0}} = 3$

Step 9.2.2

Simplify the denominator.

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Step 9.2.2.1

Multiply $2$ by $0$ .

$1 + \frac{C}{e^{0}} = 3$

Step 9.2.2.2

Anything raised to $0$ is $1$ .

$1 + \frac{C}{1} = 3$

Step 9.2.3

Divide $C$ by $1$ .

$1 + C = 3$

Step 9.3

Move all terms not containing $C$ to the right side of the equation.

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Step 9.3.1

Subtract $1$ from both sides of the equation.

$C = 3 - 1$

Step 9.3.2

Subtract $1$ from $3$ .

$C = 2$

Step 10

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

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Step 10.1

Substitute $2$ for $C$ .

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