Calculus Examples

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

Step 1

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

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Step 1.1

Set up the integration.

$e^{\int x d x}$

Step 1.2

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

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

Step 1.3

Remove the constant of integration.

$e^{\frac{1}{2} x^{2}}$

Step 1.4

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

$e^{\frac{x^{2}}{2}}$

Step 2

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

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Step 2.1

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

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

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

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

Step 3

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

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

Step 4

Set up an integral on each side.

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

Step 5

Integrate the left side.

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

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.

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

Step 6.2

Let $u_{2} = e^{\frac{x^{2}}{2}}$ . Then $d u_{2} = x e^{\frac{x^{2}}{2}} d x$ , so $\frac{1}{x e^{\frac{x^{2}}{2}}} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 6.2.1

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

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Step 6.2.1.1

Differentiate $e^{\frac{x^{2}}{2}}$ .

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

Step 6.2.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = \frac{x^{2}}{2}$ .

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Step 6.2.1.2.1

To apply the Chain Rule, set $u_{1}$ as $\frac{x^{2}}{2}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [\frac{x^{2}}{2}]$

Step 6.2.1.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [\frac{x^{2}}{2}]$

Step 6.2.1.2.3

Replace all occurrences of $u_{1}$ with $\frac{x^{2}}{2}$ .

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

Step 6.2.1.3

Differentiate.

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Step 6.2.1.3.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x^{2}]$ .

$e^{\frac{x^{2}}{2}} (\frac{1}{2} \frac{d}{d x} [x^{2}])$

Step 6.2.1.3.2

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

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

Step 6.2.1.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{e^{\frac{x^{2}}{2}}}{2} (2 x)$

Step 6.2.1.3.4

Simplify terms.

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Step 6.2.1.3.4.1

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

$\frac{2 e^{\frac{x^{2}}{2}}}{2} x$

Step 6.2.1.3.4.2

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

$\frac{2 e^{\frac{x^{2}}{2}} x}{2}$

Step 6.2.1.3.4.3

Cancel the common factor of $2$ .

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Step 6.2.1.3.4.3.1

Cancel the common factor.

$\frac{2 e^{\frac{x^{2}}{2}} x}{2}$

Step 6.2.1.3.4.3.2

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

$e^{\frac{x^{2}}{2}} x$

Step 6.2.1.3.4.4

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

$x e^{\frac{x^{2}}{2}}$

Step 6.2.2

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

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

Step 6.3

Apply the constant rule.

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

Step 6.4

Simplify the answer.

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Step 6.4.1

Simplify.

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

Step 6.4.2

Replace all occurrences of $u_{2}$ with $e^{\frac{x^{2}}{2}}$ .

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

Step 6.4.3

Reorder terms.

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

Step 7

Divide each term in $e^{\frac{x^{2}}{2}} y = 2 e^{\frac{1}{2} x^{2}} + C$ by $e^{\frac{x^{2}}{2}}$ and simplify.

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Step 7.1

Divide each term in $e^{\frac{x^{2}}{2}} y = 2 e^{\frac{1}{2} x^{2}} + C$ by $e^{\frac{x^{2}}{2}}$ .

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

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $y$ by $1$ .

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

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

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Step 7.3.1.1.1

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

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

Step 7.3.1.1.2

Cancel the common factors.

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Step 7.3.1.1.2.1

Multiply by $1$ .

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

Step 7.3.1.1.2.2

Cancel the common factor.

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

Step 7.3.1.1.2.3

Rewrite the expression.

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

Step 7.3.1.1.2.4

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

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

Step 7.3.1.2

Simplify the numerator.

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Step 7.3.1.2.1

Factor $x^{2}$ out of $\frac{1}{2} x^{2} \cdot 2 - x^{2}$ .

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Step 7.3.1.2.1.1

Factor $x^{2}$ out of $\frac{1}{2} x^{2} \cdot 2$ .

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

Step 7.3.1.2.1.2

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

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

Step 7.3.1.2.1.3

Factor $x^{2}$ out of $x^{2} (\frac{1}{2} \cdot 2) + x^{2} \cdot - 1$ .

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

Step 7.3.1.2.2

Cancel the common factor of $2$ .

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Step 7.3.1.2.2.1

Cancel the common factor.

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

Step 7.3.1.2.2.2

Rewrite the expression.

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

Step 7.3.1.2.3

Subtract $1$ from $1$ .

$y = 2 e^{\frac{x^{2} \cdot 0}{2}} + \frac{C}{e^{\frac{x^{2}}{2}}}$

Step 7.3.1.3

Multiply $x^{2}$ by $0$ .

$y = 2 e^{\frac{0}{2}} + \frac{C}{e^{\frac{x^{2}}{2}}}$

Step 7.3.1.4

Divide $0$ by $2$ .

$y = 2 e^{0} + \frac{C}{e^{\frac{x^{2}}{2}}}$

Step 7.3.1.5

Anything raised to $0$ is $1$ .

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

Step 7.3.1.6

Multiply $2$ by $1$ .

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