Calculus Examples

$\frac{d y}{d x} + 3 y = 6 e^{5 x}$ ; with $y (0) = 5$

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 3$ .

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Step 1.1

Set up the integration.

$e^{\int 3 d x}$

Step 1.2

Apply the constant rule.

$e^{3 x + C}$

Step 1.3

Remove the constant of integration.

$e^{3 x}$

Step 2

Multiply each term by the integrating factor $e^{3 x}$ .

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Step 2.1

Multiply each term by $e^{3 x}$ .

$e^{3 x} \frac{d y}{d x} + e^{3 x} (3 y) = e^{3 x} (6 e^{5 x})$

Step 2.2

Rewrite using the commutative property of multiplication.

$e^{3 x} \frac{d y}{d x} + 3 e^{3 x} y = e^{3 x} (6 e^{5 x})$

Step 2.3

Rewrite using the commutative property of multiplication.

$e^{3 x} \frac{d y}{d x} + 3 e^{3 x} y = 6 e^{3 x} e^{5 x}$

Step 2.4

Multiply $e^{3 x}$ by $e^{5 x}$ by adding the exponents.

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Step 2.4.1

Move $e^{5 x}$ .

$e^{3 x} \frac{d y}{d x} + 3 e^{3 x} y = 6 (e^{5 x} e^{3 x})$

Step 2.4.2

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

$e^{3 x} \frac{d y}{d x} + 3 e^{3 x} y = 6 e^{5 x + 3 x}$

Step 2.4.3

Add $5 x$ and $3 x$ .

$e^{3 x} \frac{d y}{d x} + 3 e^{3 x} y = 6 e^{8 x}$

Step 2.5

Reorder factors in $e^{3 x} \frac{d y}{d x} + 3 e^{3 x} y = 6 e^{8 x}$ .

$e^{3 x} \frac{d y}{d x} + 3 y e^{3 x} = 6 e^{8 x}$

Step 3

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

$\frac{d}{d x} [e^{3 x} y] = 6 e^{8 x}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{3 x} y] d x = \int 6 e^{8 x} d x$

Step 5

Integrate the left side.

$e^{3 x} y = \int 6 e^{8 x} d x$

Step 6

Integrate the right side.

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Step 6.1

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$e^{3 x} y = 6 \int e^{8 x} d x$

Step 6.2

Let $u = 8 x$ . Then $d u = 8 d x$ , so $\frac{1}{8} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.2.1

Let $u = 8 x$ . Find $\frac{d u}{d x}$ .

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Step 6.2.1.1

Differentiate $8 x$ .

$\frac{d}{d x} [8 x]$

Step 6.2.1.2

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$8 \frac{d}{d x} [x]$

Step 6.2.1.3

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

$8 \cdot 1$

Step 6.2.1.4

Multiply $8$ by $1$ .

$8$

Step 6.2.2

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

$e^{3 x} y = 6 \int e^{u} \frac{1}{8} d u$

Step 6.3

Combine $e^{u}$ and $\frac{1}{8}$ .

$e^{3 x} y = 6 \int \frac{e^{u}}{8} d u$

Step 6.4

Since $\frac{1}{8}$ is constant with respect to $u$ , move $\frac{1}{8}$ out of the integral.

$e^{3 x} y = 6 (\frac{1}{8} \int e^{u} d u)$

Step 6.5

Simplify.

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Step 6.5.1

Combine $\frac{1}{8}$ and $6$ .

$e^{3 x} y = \frac{6}{8} \int e^{u} d u$

Step 6.5.2

Cancel the common factor of $6$ and $8$ .

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Step 6.5.2.1

Factor $2$ out of $6$ .

$e^{3 x} y = \frac{2 (3)}{8} \int e^{u} d u$

Step 6.5.2.2

Cancel the common factors.

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Step 6.5.2.2.1

Factor $2$ out of $8$ .

$e^{3 x} y = \frac{2 \cdot 3}{2 \cdot 4} \int e^{u} d u$

Step 6.5.2.2.2

Cancel the common factor.

$e^{3 x} y = \frac{2 \cdot 3}{2 \cdot 4} \int e^{u} d u$

Step 6.5.2.2.3

Rewrite the expression.

$e^{3 x} y = \frac{3}{4} \int e^{u} d u$

Step 6.6

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

$e^{3 x} y = \frac{3}{4} (e^{u} + C)$

Step 6.7

Simplify.

$e^{3 x} y = \frac{3}{4} e^{u} + C$

Step 6.8

Replace all occurrences of $u$ with $8 x$ .

$e^{3 x} y = \frac{3}{4} e^{8 x} + C$

Step 7

Divide each term in $e^{3 x} y = \frac{3}{4} e^{8 x} + C$ by $e^{3 x}$ and simplify.

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Step 7.1

Divide each term in $e^{3 x} y = \frac{3}{4} e^{8 x} + C$ by $e^{3 x}$ .

$\frac{e^{3 x} y}{e^{3 x}} = \frac{\frac{3}{4} e^{8 x}}{e^{3 x}} + \frac{C}{e^{3 x}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $e^{3 x}$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{e^{3 x} y}{e^{3 x}} = \frac{\frac{3}{4} e^{8 x}}{e^{3 x}} + \frac{C}{e^{3 x}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{3}{4} e^{8 x}}{e^{3 x}} + \frac{C}{e^{3 x}}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Cancel the common factor of $e^{8 x}$ and $e^{3 x}$ .

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Step 7.3.1.1.1

Factor $e^{3 x}$ out of $\frac{3}{4} e^{8 x}$ .

$y = \frac{e^{3 x} (\frac{3}{4} e^{5 x})}{e^{3 x}} + \frac{C}{e^{3 x}}$

Step 7.3.1.1.2

Cancel the common factors.

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Step 7.3.1.1.2.1

Multiply by $1$ .

$y = \frac{e^{3 x} (\frac{3}{4} e^{5 x})}{e^{3 x} \cdot 1} + \frac{C}{e^{3 x}}$

Step 7.3.1.1.2.2

Cancel the common factor.

$y = \frac{e^{3 x} (\frac{3}{4} e^{5 x})}{e^{3 x} \cdot 1} + \frac{C}{e^{3 x}}$

Step 7.3.1.1.2.3

Rewrite the expression.

$y = \frac{\frac{3}{4} e^{5 x}}{1} + \frac{C}{e^{3 x}}$

Step 7.3.1.1.2.4

Divide $\frac{3}{4} e^{5 x}$ by $1$ .

$y = \frac{3}{4} e^{5 x} + \frac{C}{e^{3 x}}$

Step 7.3.1.2

Combine $\frac{3}{4}$ and $e^{5 x}$ .

$y = \frac{3 e^{5 x}}{4} + \frac{C}{e^{3 x}}$

Step 8

Use the initial condition to find the value of $C$ by substituting $0$ for $x$ and $5$ for $y$ in $y = \frac{3 e^{5 x}}{4} + \frac{C}{e^{3 x}}$ .

$5 = \frac{3 e^{5 \cdot 0}}{4} + \frac{C}{e^{3 \cdot 0}}$

Step 9

Solve for $C$ .

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Step 9.1

Rewrite the equation as $\frac{3 e^{5 \cdot 0}}{4} + \frac{C}{e^{3 \cdot 0}} = 5$ .

$\frac{3 e^{5 \cdot 0}}{4} + \frac{C}{e^{3 \cdot 0}} = 5$

Step 9.2

Simplify $\frac{3 e^{5 \cdot 0}}{4} + \frac{C}{e^{3 \cdot 0}}$ .

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Step 9.2.1

Multiply $5$ by $0$ .

$\frac{3 e^{0}}{4} + \frac{C}{e^{3 \cdot 0}} = 5$

Step 9.2.2

Simplify each term.

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Step 9.2.2.1

Anything raised to $0$ is $1$ .

$\frac{3 \cdot 1}{4} + \frac{C}{e^{3 \cdot 0}} = 5$

Step 9.2.2.2

Multiply $3$ by $1$ .

$\frac{3}{4} + \frac{C}{e^{3 \cdot 0}} = 5$

Step 9.2.2.3

Simplify the denominator.

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Step 9.2.2.3.1

Multiply $3$ by $0$ .

$\frac{3}{4} + \frac{C}{e^{0}} = 5$

Step 9.2.2.3.2

Anything raised to $0$ is $1$ .

$\frac{3}{4} + \frac{C}{1} = 5$

Step 9.2.2.4

Divide $C$ by $1$ .

$\frac{3}{4} + C = 5$

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 $\frac{3}{4}$ from both sides of the equation.

$C = 5 - \frac{3}{4}$

Step 9.3.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$C = 5 \cdot \frac{4}{4} - \frac{3}{4}$

Step 9.3.3

Combine $5$ and $\frac{4}{4}$ .

$C = \frac{5 \cdot 4}{4} - \frac{3}{4}$

Step 9.3.4

Combine the numerators over the common denominator.

$C = \frac{5 \cdot 4 - 3}{4}$

Step 9.3.5

Simplify the numerator.

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Step 9.3.5.1

Multiply $5$ by $4$ .

$C = \frac{20 - 3}{4}$

Step 9.3.5.2

Subtract $3$ from $20$ .

$C = \frac{17}{4}$

Step 10

Substitute $\frac{17}{4}$ for $C$ in $y = \frac{3 e^{5 x}}{4} + \frac{C}{e^{3 x}}$ and simplify.

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Step 10.1

Substitute $\frac{17}{4}$ for $C$ .

$y = \frac{3 e^{5 x}}{4} + \frac{\frac{17}{4}}{e^{3 x}}$

Step 10.2

Simplify each term.

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Step 10.2.1

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{3 e^{5 x}}{4} + \frac{17}{4} \cdot \frac{1}{e^{3 x}}$

Step 10.2.2

Combine.

$y = \frac{3 e^{5 x}}{4} + \frac{17 \cdot 1}{4 e^{3 x}}$

Step 10.2.3

Multiply $17$ by $1$ .

$y = \frac{3 e^{5 x}}{4} + \frac{17}{4 e^{3 x}}$