Calculus Examples

$\frac{d y}{d x} + 4 y = 2 x$

Step 1

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

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Step 1.1

Set up the integration.

$e^{\int 4 d x}$

Step 1.2

Apply the constant rule.

$e^{4 x + C}$

Step 1.3

Remove the constant of integration.

$e^{4 x}$

Step 2

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

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Step 2.1

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

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

Rewrite using the commutative property of multiplication.

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

Step 2.4

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

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

Step 3

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

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

Step 4

Set up an integral on each side.

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

Step 5

Integrate the left side.

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

Step 6

Integrate the right side.

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Step 6.1

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

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

Step 6.2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{4 x}$ .

$e^{4 x} y = 2 (x (\frac{1}{4} e^{4 x}) - \int \frac{1}{4} e^{4 x} d x)$

Step 6.3

Simplify.

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Step 6.3.1

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

$e^{4 x} y = 2 (x \frac{e^{4 x}}{4} - \int \frac{1}{4} e^{4 x} d x)$

Step 6.3.2

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

$e^{4 x} y = 2 (\frac{x e^{4 x}}{4} - \int \frac{1}{4} e^{4 x} d x)$

Step 6.3.3

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

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

Step 6.4

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

$e^{4 x} y = 2 (\frac{x e^{4 x}}{4} - (\frac{1}{4} \int e^{4 x} d x))$

Step 6.5

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

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Step 6.5.1

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

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Step 6.5.1.1

Differentiate $4 x$ .

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

Step 6.5.1.2

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

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

Step 6.5.1.3

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

$4 \cdot 1$

Step 6.5.1.4

Multiply $4$ by $1$ .

$4$

Step 6.5.2

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

$e^{4 x} y = 2 (\frac{x e^{4 x}}{4} - \frac{1}{4} \int e^{u} \frac{1}{4} d u)$

Step 6.6

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

$e^{4 x} y = 2 (\frac{x e^{4 x}}{4} - \frac{1}{4} \int \frac{e^{u}}{4} d u)$

Step 6.7

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

$e^{4 x} y = 2 (\frac{x e^{4 x}}{4} - \frac{1}{4} (\frac{1}{4} \int e^{u} d u))$

Step 6.8

Simplify.

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Step 6.8.1

Multiply $\frac{1}{4}$ by $\frac{1}{4}$ .

$e^{4 x} y = 2 (\frac{x e^{4 x}}{4} - \frac{1}{4 \cdot 4} \int e^{u} d u)$

Step 6.8.2

Multiply $4$ by $4$ .

$e^{4 x} y = 2 (\frac{x e^{4 x}}{4} - \frac{1}{16} \int e^{u} d u)$

Step 6.9

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

$e^{4 x} y = 2 (\frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u} + C))$

Step 6.10

Rewrite $2 (\frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u} + C))$ as $2 (\frac{1}{4} x e^{4 x} - \frac{1}{16} e^{u}) + C$ .

$e^{4 x} y = 2 (\frac{1}{4} x e^{4 x} - \frac{1}{16} e^{u}) + C$

Step 6.11

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

$e^{4 x} y = 2 (\frac{1}{4} x e^{4 x} - \frac{1}{16} e^{4 x}) + C$

Step 6.12

Simplify.

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Step 6.12.1

Simplify each term.

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Step 6.12.1.1

Combine $\frac{1}{4}$ and $x$ .

$e^{4 x} y = 2 (\frac{x}{4} e^{4 x} - \frac{1}{16} e^{4 x}) + C$

Step 6.12.1.2

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

$e^{4 x} y = 2 (\frac{x e^{4 x}}{4} - \frac{1}{16} e^{4 x}) + C$

Step 6.12.1.3

Combine $e^{4 x}$ and $\frac{1}{16}$ .

$e^{4 x} y = 2 (\frac{x e^{4 x}}{4} - \frac{e^{4 x}}{16}) + C$

Step 6.12.2

Apply the distributive property.

$e^{4 x} y = 2 \frac{x e^{4 x}}{4} + 2 (- \frac{e^{4 x}}{16}) + C$

Step 6.12.3

Cancel the common factor of $2$ .

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Step 6.12.3.1

Factor $2$ out of $4$ .

$e^{4 x} y = 2 \frac{x e^{4 x}}{2 (2)} + 2 (- \frac{e^{4 x}}{16}) + C$

Step 6.12.3.2

Cancel the common factor.

$e^{4 x} y = 2 \frac{x e^{4 x}}{2 \cdot 2} + 2 (- \frac{e^{4 x}}{16}) + C$

Step 6.12.3.3

Rewrite the expression.

$e^{4 x} y = \frac{x e^{4 x}}{2} + 2 (- \frac{e^{4 x}}{16}) + C$

Step 6.12.4

Cancel the common factor of $2$ .

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Step 6.12.4.1

Move the leading negative in $- \frac{e^{4 x}}{16}$ into the numerator.

$e^{4 x} y = \frac{x e^{4 x}}{2} + 2 \frac{- e^{4 x}}{16} + C$

Step 6.12.4.2

Factor $2$ out of $16$ .

$e^{4 x} y = \frac{x e^{4 x}}{2} + 2 \frac{- e^{4 x}}{2 (8)} + C$

Step 6.12.4.3

Cancel the common factor.

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

Step 6.12.4.4

Rewrite the expression.

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

Step 6.12.5

Move the negative in front of the fraction.

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

Step 6.13

Reorder terms.

$e^{4 x} y = \frac{1}{2} x e^{4 x} - \frac{1}{8} e^{4 x} + C$

Step 7

Divide each term in $e^{4 x} y = \frac{1}{2} x e^{4 x} - \frac{1}{8} e^{4 x} + C$ by $e^{4 x}$ and simplify.

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Step 7.1

Divide each term in $e^{4 x} y = \frac{1}{2} x e^{4 x} - \frac{1}{8} e^{4 x} + C$ by $e^{4 x}$ .

$\frac{e^{4 x} y}{e^{4 x}} = \frac{\frac{1}{2} x e^{4 x}}{e^{4 x}} + \frac{- \frac{1}{8} e^{4 x}}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

$\frac{e^{4 x} y}{e^{4 x}} = \frac{\frac{1}{2} x e^{4 x}}{e^{4 x}} + \frac{- \frac{1}{8} e^{4 x}}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{2} x e^{4 x}}{e^{4 x}} + \frac{- \frac{1}{8} e^{4 x}}{e^{4 x}} + \frac{C}{e^{4 x}}$

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 $e^{4 x}$ .

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Step 7.3.1.1.1

Cancel the common factor.

$y = \frac{\frac{1}{2} x e^{4 x}}{e^{4 x}} + \frac{- \frac{1}{8} e^{4 x}}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.3.1.1.2

Divide $\frac{1}{2} x$ by $1$ .

$y = \frac{1}{2} x + \frac{- \frac{1}{8} e^{4 x}}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.3.1.2

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

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Step 7.3.1.2.1

Cancel the common factor.

$y = \frac{1}{2} x + \frac{- \frac{1}{8} e^{4 x}}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.3.1.2.2

Divide $- \frac{1}{8}$ by $1$ .

$y = \frac{1}{2} x - \frac{1}{8} + \frac{C}{e^{4 x}}$

Step 7.3.2

Subtract $\frac{1}{8}$ from $\frac{1}{2} x$ .

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Step 7.3.2.1

Reorder $\frac{1}{2}$ and $x$ .

$y = x \frac{1}{2} - \frac{1}{8} + \frac{C}{e^{4 x}}$

Step 7.3.2.2

To write $x \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

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

Step 7.3.2.3

Combine $x \frac{1}{2}$ and $\frac{8}{8}$ .

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

Step 7.3.2.4

Combine the numerators over the common denominator.

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

Step 7.3.3

Simplify the numerator.

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Step 7.3.3.1

Combine $x$ and $\frac{1}{2}$ .

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

Step 7.3.3.2

Cancel the common factor of $2$ .

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Step 7.3.3.2.1

Factor $2$ out of $8$ .

$y = \frac{\frac{x}{2} \cdot (2 (4)) - 1}{8} + \frac{C}{e^{4 x}}$

Step 7.3.3.2.2

Cancel the common factor.

$y = \frac{\frac{x}{2} \cdot (2 \cdot 4) - 1}{8} + \frac{C}{e^{4 x}}$

Step 7.3.3.2.3

Rewrite the expression.

$y = \frac{x \cdot 4 - 1}{8} + \frac{C}{e^{4 x}}$

Step 7.3.3.3

Move $4$ to the left of $x$ .

$y = \frac{4 x - 1}{8} + \frac{C}{e^{4 x}}$

Step 7.3.4

To write $\frac{4 x - 1}{8}$ as a fraction with a common denominator, multiply by $\frac{e^{4 x}}{e^{4 x}}$ .

$y = \frac{4 x - 1}{8} \cdot \frac{e^{4 x}}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.3.5

To write $\frac{C}{e^{4 x}}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = \frac{4 x - 1}{8} \cdot \frac{e^{4 x}}{e^{4 x}} + \frac{C}{e^{4 x}} \cdot \frac{8}{8}$

Step 7.3.6

Write each expression with a common denominator of $8 e^{4 x}$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.3.6.1

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

$y = \frac{(4 x - 1) e^{4 x}}{8 e^{4 x}} + \frac{C}{e^{4 x}} \cdot \frac{8}{8}$

Step 7.3.6.2

Multiply $\frac{C}{e^{4 x}}$ by $\frac{8}{8}$ .

$y = \frac{(4 x - 1) e^{4 x}}{8 e^{4 x}} + \frac{C \cdot 8}{e^{4 x} \cdot 8}$

Step 7.3.6.3

Reorder the factors of $e^{4 x} \cdot 8$ .

$y = \frac{(4 x - 1) e^{4 x}}{8 e^{4 x}} + \frac{C \cdot 8}{8 e^{4 x}}$

Step 7.3.7

Combine the numerators over the common denominator.

$y = \frac{(4 x - 1) e^{4 x} + C \cdot 8}{8 e^{4 x}}$

Step 7.3.8

Simplify the numerator.

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Step 7.3.8.1

Apply the distributive property.

$y = \frac{4 x e^{4 x} - 1 e^{4 x} + C \cdot 8}{8 e^{4 x}}$

Step 7.3.8.2

Rewrite $- 1 e^{4 x}$ as $- e^{4 x}$ .

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

Step 7.3.8.3

Move $8$ to the left of $C$ .

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