Calculus Examples

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

Step 1

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

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Step 1.1

Set up the integration.

$e^{\int - 2 d x}$

Step 1.2

Apply the constant rule.

$e^{- 2 x + C}$

Step 1.3

Remove the constant of integration.

$e^{- 2 x}$

Step 2

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

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Step 2.1

Multiply each term by $e^{- 2 x}$ .

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

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

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

Step 3

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

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

Step 4

Set up an integral on each side.

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

Step 5

Integrate the left side.

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

Step 6

Integrate the right side.

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Step 6.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{- 2 x}$ .

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

Step 6.2

Simplify.

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Step 6.2.1

Combine $e^{- 2 x}$ and $\frac{1}{2}$ .

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

Step 6.2.2

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

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

Step 6.3

Since $- \frac{1}{2}$ is constant with respect to $x$ , move $- \frac{1}{2}$ out of the integral.

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

Step 6.4

Simplify.

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Step 6.4.1

Multiply $- 1$ by $- 1$ .

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

Step 6.4.2

Multiply $\frac{1}{2}$ by $1$ .

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

Step 6.5

Let $u = - 2 x$ . Then $d u = - 2 d x$ , so $- \frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.5.1

Let $u = - 2 x$ . Find $\frac{d u}{d x}$ .

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Step 6.5.1.1

Differentiate $- 2 x$ .

$\frac{d}{d x} [- 2 x]$

Step 6.5.1.2

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$- 2 \frac{d}{d x} [x]$

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 $n = 1$ .

$- 2 \cdot 1$

Step 6.5.1.4

Multiply $- 2$ by $1$ .

$- 2$

Step 6.5.2

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

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

Step 6.6

Simplify.

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Step 6.6.1

Move the negative in front of the fraction.

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

Step 6.6.2

Combine $e^{u}$ and $\frac{1}{2}$ .

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

Step 6.7

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

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

Step 6.8

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 6.9

Simplify.

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Step 6.9.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

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

Step 6.9.2

Multiply $2$ by $2$ .

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

Step 6.10

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

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

Step 6.11

Simplify.

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Step 6.11.1

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

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

Step 6.11.2

Simplify.

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Step 6.11.2.1

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

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

Step 6.11.2.2

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

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

Step 6.12

Replace all occurrences of $u$ with $- 2 x$ .

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

Step 6.13

Combine $e^{- 2 x}$ and $\frac{1}{4}$ .

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

Step 6.14

Reorder terms.

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

Step 7

Solve for $y$ .

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Step 7.1

Simplify.

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Step 7.1.1

Combine $e^{- 2 x}$ and $\frac{1}{2}$ .

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

Step 7.1.2

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

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

Step 7.1.3

Combine $e^{- 2 x}$ and $\frac{1}{4}$ .

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

Step 7.2

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

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Step 7.2.1

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

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

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

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

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Step 7.2.2.1.1

Cancel the common factor.

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

Step 7.2.2.1.2

Divide $y$ by $1$ .

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

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Simplify each term.

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Step 7.2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.2.3.1.2

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

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

Step 7.2.3.1.3

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

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Step 7.2.3.1.3.1

Cancel the common factor.

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

Step 7.2.3.1.3.2

Rewrite the expression.

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

Step 7.2.3.1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.2.3.1.5

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

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

Step 7.2.3.1.6

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

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Step 7.2.3.1.6.1

Cancel the common factor.

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

Step 7.2.3.1.6.2

Rewrite the expression.

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