Calculus Examples

$y' = x + y$

Step 1

Rewrite the differential equation.

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

Step 2

Subtract $y$ from both sides of the equation.

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

Step 3

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

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Step 3.1

Set up the integration.

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

Step 3.2

Apply the constant rule.

$e^{- x + C}$

Step 3.3

Remove the constant of integration.

$e^{- x}$

Step 4

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

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Step 4.1

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

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

Step 4.2

Rewrite using the commutative property of multiplication.

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

Step 4.3

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

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

Step 5

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

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

Step 6

Set up an integral on each side.

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

Step 7

Integrate the left side.

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

Step 8

Integrate the right side.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

Simplify.

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Step 8.3.1

Multiply $- 1$ by $- 1$ .

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

Step 8.3.2

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

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

Step 8.4

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

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Step 8.4.1

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

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Step 8.4.1.1

Differentiate $- x$ .

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

Step 8.4.1.2

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 8.4.1.3

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 8.4.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 8.4.2

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

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

Step 8.5

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

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

Step 8.6

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

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

Step 8.7

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

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

Step 8.8

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

$e^{- x} y = - x e^{- x} - e^{- x} + C$

Step 9

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

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Step 9.1

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

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

Step 9.2

Simplify the left side.

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Step 9.2.1

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

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Step 9.2.1.1

Cancel the common factor.

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

Step 9.2.1.2

Divide $y$ by $1$ .

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

Step 9.3

Simplify the right side.

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Step 9.3.1

Simplify each term.

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Step 9.3.1.1

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

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Step 9.3.1.1.1

Cancel the common factor.

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

Step 9.3.1.1.2

Divide $- x$ by $1$ .

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

Step 9.3.1.2

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

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Step 9.3.1.2.1

Cancel the common factor.

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

Step 9.3.1.2.2

Divide $- 1$ by $1$ .

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