Calculus Examples

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

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = - 1$ .

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Step 1.1

Set up the integration.

$e^{\int - 1 d x}$

Step 1.2

Apply the constant rule.

$e^{- x + C}$

Step 1.3

Remove the constant of integration.

$e^{- x}$

Step 2

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

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Step 2.1

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

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

Rewrite using the commutative property of multiplication.

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

Step 2.4

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

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

Step 3

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

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

Step 4

Set up an integral on each side.

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

Step 5

Integrate the left side.

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

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.

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Simplify.

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Step 6.4.1

Multiply $- 1$ by $- 1$ .

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

Step 6.4.2

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

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

Step 6.5

Let $u = - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.5.1

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

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Step 6.5.1.1

Differentiate $- x$ .

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

Step 6.5.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

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 $n = 1$ .

$- 1 \cdot 1$

Step 6.5.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 6.5.2

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

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

Step 6.6

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

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

Step 6.7

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

$e^{- x} y = 2 (x (- e^{- x}) - (e^{u} + C))$

Step 6.8

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

$e^{- x} y = 2 (- x e^{- x} - e^{u}) + C$

Step 6.9

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

$e^{- x} y = 2 (- x e^{- x} - e^{- x}) + C$

Step 7

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

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Step 7.1

Divide each term in $e^{- x} y = 2 (- x e^{- x} - e^{- x}) + C$ by $e^{- x}$ .

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

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $y$ by $1$ .

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

Step 7.3

Simplify the right side.

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Step 7.3.1

Combine the numerators over the common denominator.

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

Step 7.3.2

Simplify the numerator.

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Step 7.3.2.1

Apply the distributive property.

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

Step 7.3.2.2

Multiply $- 1$ by $2$ .

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

Step 7.3.2.3

Multiply $- 1$ by $2$ .

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

Step 7.3.3

Simplify with factoring out.

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Step 7.3.3.1

Factor $- 1$ out of $- 2 x e^{- x}$ .

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

Step 7.3.3.2

Factor $- 1$ out of $- 2 e^{- x}$ .

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

Step 7.3.3.3

Factor $- 1$ out of $- (2 x e^{- x}) - (2 e^{- x})$ .

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

Step 7.3.3.4

Factor $- 1$ out of $C$ .

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

Step 7.3.3.5

Factor $- 1$ out of $- (2 x e^{- x} + 2 e^{- x}) - 1 (- C)$ .

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

Step 7.3.3.6

Simplify the expression.

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Step 7.3.3.6.1

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

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

Step 7.3.3.6.2

Move the negative in front of the fraction.

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