Calculus Examples

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

Step 1

Subtract $y$ from both sides of the equation.

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

Step 2

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

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Step 2.1

Set up the integration.

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

Step 2.2

Apply the constant rule.

$e^{- x + C}$

Step 2.3

Remove the constant of integration.

$e^{- x}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Simplify each term.

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Step 3.3.1

Rewrite using the commutative property of multiplication.

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Move $- 1$ to the left of $e^{- x}$ .

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

Step 3.3.5

Rewrite $- 1 e^{- x}$ as $- e^{- x}$ .

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

Step 3.4

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

Simplify.

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Step 7.5.1

Multiply $- 1$ by $- 1$ .

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

Step 7.5.2

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

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

Step 7.6

Let $u_{1} = - x$ . Then $d u_{1} = - d x$ , so $- d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 7.6.1

Let $u_{1} = - x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 7.6.1.1

Differentiate $- x$ .

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

Step 7.6.1.2

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 7.6.1.3

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 7.6.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 7.6.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

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

Step 7.7

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

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

Step 7.8

The integral of $e^{u_{1}}$ with respect to $u_{1}$ is $e^{u_{1}}$ .

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

Step 7.9

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

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

Step 7.10

Let $u_{2} = - x$ . Then $d u_{2} = - d x$ , so $- d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 7.10.1

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

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Step 7.10.1.1

Differentiate $- x$ .

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

Step 7.10.1.2

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 7.10.1.3

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 7.10.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 7.10.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

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

Step 7.11

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

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

Step 7.12

Simplify.

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Step 7.12.1

Multiply $- 1$ by $- 1$ .

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

Step 7.12.2

Multiply $\int e^{u_{2}} d u_{2}$ by $1$ .

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

Step 7.13

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

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

Step 7.14

Simplify.

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

Step 7.15

Substitute back in for each integration substitution variable.

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Step 7.15.1

Replace all occurrences of $u_{1}$ with $- x$ .

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

Step 7.15.2

Replace all occurrences of $u_{2}$ with $- x$ .

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

Step 7.16

Simplify.

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Step 7.16.1

Apply the distributive property.

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

Step 7.16.2

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

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Step 7.16.2.1

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

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

Step 7.16.2.2

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

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

Step 7.16.3

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

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

Step 7.16.4

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

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

Step 7.17

Reorder terms.

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

Step 8

Solve for $y$ .

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Step 8.1

Simplify.

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Step 8.1.1

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

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

Step 8.1.2

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

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

Step 8.1.3

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

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

Step 8.2

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

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Step 8.2.1

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

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

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

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

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Step 8.2.2.1.1

Cancel the common factor.

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

Step 8.2.2.1.2

Divide $y$ by $1$ .

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

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Combine the numerators over the common denominator.

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

Step 8.2.3.2

To write $e^{- x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 8.2.3.3

Simplify terms.

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Step 8.2.3.3.1

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

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

Step 8.2.3.3.2

Combine the numerators over the common denominator.

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

Step 8.2.3.3.3

Combine the numerators over the common denominator.

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

Step 8.2.3.4

Move $2$ to the left of $e^{- x}$ .

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

Step 8.2.3.5

Simplify terms.

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Step 8.2.3.5.1

Add $- e^{- x}$ and $2 e^{- x}$ .

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

Step 8.2.3.5.2

Reorder factors in $- e^{- x} x + e^{- x}$ .

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

Step 8.2.3.5.3

Factor $e^{- x}$ out of $- x e^{- x} + e^{- x}$ .

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Step 8.2.3.5.3.1

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

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

Step 8.2.3.5.3.2

Multiply by $1$ .

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

Step 8.2.3.5.3.3

Factor $e^{- x}$ out of $e^{- x} (- x) + e^{- x} \cdot 1$ .

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

Step 8.2.3.6

To write $C$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 8.2.3.7

Simplify terms.

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Step 8.2.3.7.1

Combine $C$ and $\frac{2}{2}$ .

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

Step 8.2.3.7.2

Combine the numerators over the common denominator.

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

Step 8.2.3.8

Simplify the numerator.

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Step 8.2.3.8.1

Move $2$ to the left of $C$ .

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

Step 8.2.3.8.2

Apply the distributive property.

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

Step 8.2.3.8.3

Rewrite using the commutative property of multiplication.

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

Step 8.2.3.8.4

Multiply $e^{- x}$ by $1$ .

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

Step 8.2.3.9

Reorder factors in $\frac{2 C - e^{- x} x + e^{- x}}{2}$ .

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

Step 8.2.3.10

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2.3.11

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

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