Calculus Examples

$\frac{d y}{d x} + 4 y - e^{- x} = 0$ , $y (0) = \frac{4}{3}$

Step 1

Add $e^{- x}$ to both sides of the equation.

$\frac{d y}{d x} + 4 y = e^{- x}$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 4$ .

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Step 2.1

Set up the integration.

$e^{\int 4 d x}$

Step 2.2

Apply the constant rule.

$e^{4 x + C}$

Step 2.3

Remove the constant of integration.

$e^{4 x}$

Step 3

Multiply each term by the integrating factor $e^{4 x}$ .

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Step 3.1

Multiply each term by $e^{4 x}$ .

$e^{4 x} \frac{d y}{d x} + e^{4 x} (4 y) = e^{4 x} e^{- x}$

Step 3.2

Rewrite using the commutative property of multiplication.

$e^{4 x} \frac{d y}{d x} + 4 e^{4 x} y = e^{4 x} e^{- x}$

Step 3.3

Multiply $e^{4 x}$ by $e^{- x}$ 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.

$e^{4 x} \frac{d y}{d x} + 4 e^{4 x} y = e^{4 x - x}$

Step 3.3.2

Subtract $x$ from $4 x$ .

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

Step 3.4

Reorder factors in $e^{4 x} \frac{d y}{d x} + 4 e^{4 x} y = e^{3 x}$ .

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

Step 4

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

$\frac{d}{d x} [e^{4 x} y] = e^{3 x}$

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

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

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Step 7.1.1

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

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Step 7.1.1.1

Differentiate $3 x$ .

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

Step 7.1.1.2

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

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

Step 7.1.1.3

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

$3 \cdot 1$

Step 7.1.1.4

Multiply $3$ by $1$ .

$3$

Step 7.1.2

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

$e^{4 x} y = \int e^{u} \frac{1}{3} d u$

Step 7.2

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

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

Step 7.3

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

$e^{4 x} y = \frac{1}{3} \int e^{u} d u$

Step 7.4

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

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

Step 7.5

Simplify.

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

Step 7.6

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

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

Step 8

Divide each term in $e^{4 x} y = \frac{1}{3} e^{3 x} + C$ by $e^{4 x}$ and simplify.

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Step 8.1

Divide each term in $e^{4 x} y = \frac{1}{3} e^{3 x} + C$ by $e^{4 x}$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $e^{4 x}$ .

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

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 $e^{3 x}$ and $e^{4 x}$ .

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Step 8.3.1.1.1

Factor $e^{4 x}$ out of $\frac{1}{3} e^{3 x}$ .

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

Step 8.3.1.1.2

Cancel the common factors.

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Step 8.3.1.1.2.1

Multiply by $1$ .

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

Step 8.3.1.1.2.2

Cancel the common factor.

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

Step 8.3.1.1.2.3

Rewrite the expression.

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

Step 8.3.1.1.2.4

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

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

Step 8.3.1.2

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

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

Step 9

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

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

Step 10

Solve for $C$ .

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Step 10.1

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

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

Step 10.2

Simplify each term.

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Step 10.2.1

Anything raised to $0$ is $1$ .

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

Step 10.2.2

Simplify the denominator.

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Step 10.2.2.1

Multiply $4$ by $0$ .

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

Step 10.2.2.2

Anything raised to $0$ is $1$ .

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

Step 10.2.3

Divide $C$ by $1$ .

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

Step 10.3

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

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Step 10.3.1

Subtract $\frac{1}{3}$ from both sides of the equation.

$C = \frac{4}{3} - \frac{1}{3}$

Step 10.3.2

Combine the numerators over the common denominator.

$C = \frac{4 - 1}{3}$

Step 10.3.3

Subtract $1$ from $4$ .

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

Step 10.3.4

Divide $3$ by $3$ .

$C = 1$

Step 11

Substitute $1$ for $C$ in $y = \frac{e^{- x}}{3} + \frac{C}{e^{4 x}}$ and simplify.

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Step 11.1

Substitute $1$ for $C$ .

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