Calculus Examples

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

Step 1

Subtract $4 y$ from both sides of the equation.

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

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 $e^{- 4 x}$ .

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

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 3.4

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 4

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 5

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 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

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 7.2

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 7.3

Simplify.

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Step 7.3.1

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

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

Step 7.3.2

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 7.3.3

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

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

Step 7.4

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 7.5

Simplify.

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Step 7.5.1

Multiply $- 1$ by $- 1$ .

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

Step 7.5.2

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

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

Step 7.6

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 7.7

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 7.7.1

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

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Step 7.7.1.1

Differentiate $- 4 x$ .

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

Step 7.7.1.2

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 7.7.1.3

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 7.7.1.4

Multiply $- 4$ by $1$ .

$- 4$

Step 7.7.2

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 7.8

Simplify.

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Step 7.8.1

Move the negative in front of the fraction.

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

Step 7.8.2

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 7.9

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

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

Step 7.10

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 7.11

Simplify.

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Step 7.11.1

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 7.11.2

Multiply $4$ by $4$ .

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

Step 7.12

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 7.13

Simplify.

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Step 7.13.1

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 7.13.2

Simplify.

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Step 7.13.2.1

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

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

Step 7.13.2.2

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

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

Step 7.13.2.3

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

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

Step 7.14

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

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

Step 7.15

Simplify.

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Step 7.15.1

Apply the distributive property.

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

Step 7.15.2

Cancel the common factor of $2$ .

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Step 7.15.2.1

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

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

Step 7.15.2.2

Factor $2$ out of $4$ .

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

Step 7.15.2.3

Cancel the common factor.

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

Step 7.15.2.4

Rewrite the expression.

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

Step 7.15.3

Cancel the common factor of $2$ .

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Step 7.15.3.1

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

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

Step 7.15.3.2

Factor $2$ out of $16$ .

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

Step 7.15.3.3

Cancel the common factor.

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

Step 7.15.3.4

Rewrite the expression.

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

Step 7.15.4

Simplify each term.

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Step 7.15.4.1

Move the negative in front of the fraction.

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

Step 7.15.4.2

Move the negative in front of the fraction.

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

Step 7.15.5

Reorder factors in $- \frac{e^{- 4 x} x}{2} - \frac{e^{- 4 x}}{8}$ .

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

Step 7.16

Reorder terms.

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

Step 8

Solve for $y$ .

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Step 8.1

Simplify.

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Step 8.1.1

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

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

Step 8.1.2

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

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

Step 8.1.3

Combine $e^{- 4 x}$ and $\frac{1}{8}$ .

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

Step 8.2

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

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Step 8.2.1

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

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

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

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

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Step 8.2.2.1.1

Cancel the common factor.

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

Step 8.2.2.1.2

Divide $y$ by $1$ .

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

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Simplify each term.

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Step 8.2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2.3.1.2

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

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

Step 8.2.3.1.3

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

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Step 8.2.3.1.3.1

Cancel the common factor.

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

Step 8.2.3.1.3.2

Rewrite the expression.

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

Step 8.2.3.1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2.3.1.5

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

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

Step 8.2.3.1.6

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

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Step 8.2.3.1.6.1

Cancel the common factor.

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

Step 8.2.3.1.6.2

Rewrite the expression.

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