Calculus Examples

$\frac{d y}{d x} + y = x$ , $y (0) = 3$

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.

$e^{\int d x}$

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 $e^{x}$ .

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

Step 2.2

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 3

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

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

Step 4

Set up an integral on each side.

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

Step 5

Integrate the left side.

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

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 $d v = e^{x}$ .

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

Step 6.2

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

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

Step 6.3

Simplify.

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

Step 7

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

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Step 7.1

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

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

Step 7.2.1.2

Divide $y$ by $1$ .

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

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 $e^{x}$ .

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Step 7.3.1.1.1

Cancel the common factor.

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

Step 7.3.1.1.2

Divide $x$ by $1$ .

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

Step 7.3.1.2

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

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Step 7.3.1.2.1

Cancel the common factor.

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

Step 7.3.1.2.2

Divide $- 1$ by $1$ .

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

Step 8

Use the initial condition to find the value of $C$ by substituting $0$ for $x$ and $3$ for $y$ in $y = x - 1 + \frac{C}{e^{x}}$ .

$3 = 0 - 1 + \frac{C}{e^{0}}$

Step 9

Solve for $C$ .

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Step 9.1

Rewrite the equation as $0 - 1 + \frac{C}{e^{0}} = 3$ .

$0 - 1 + \frac{C}{e^{0}} = 3$

Step 9.2

Simplify $0 - 1 + \frac{C}{e^{0}}$ .

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Step 9.2.1

Subtract $1$ from $0$ .

$- 1 + \frac{C}{e^{0}} = 3$

Step 9.2.2

Simplify each term.

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Step 9.2.2.1

Anything raised to $0$ is $1$ .

$- 1 + \frac{C}{1} = 3$

Step 9.2.2.2

Divide $C$ by $1$ .

$- 1 + C = 3$

Step 9.3

Move all terms not containing $C$ to the right side of the equation.

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Step 9.3.1

Add $1$ to both sides of the equation.

$C = 3 + 1$

Step 9.3.2

Add $3$ and $1$ .

$C = 4$

Step 10

Substitute $4$ for $C$ in $y = x - 1 + \frac{C}{e^{x}}$ and simplify.

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Step 10.1

Substitute $4$ for $C$ .

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