Calculus Examples

$\frac{d y}{d x} - 2 x y = 6 x$

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = - 2 x$ .

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Step 1.1

Set up the integration.

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

Step 1.2

Integrate $- 2 x$ .

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Step 1.2.1

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

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

Step 1.2.2

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

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

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

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

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

Step 1.2.3.2

Simplify.

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Step 1.2.3.2.1

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

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

Step 1.2.3.2.2

Cancel the common factor of $- 2$ and $2$ .

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Step 1.2.3.2.2.1

Factor $2$ out of $- 2$ .

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

Step 1.2.3.2.2.2

Cancel the common factors.

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Step 1.2.3.2.2.2.1

Factor $2$ out of $2$ .

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

Step 1.2.3.2.2.2.2

Cancel the common factor.

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

Step 1.2.3.2.2.2.3

Rewrite the expression.

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

Step 1.2.3.2.2.2.4

Divide $- 1$ by $1$ .

$e^{- x^{2} + C}$

Step 1.3

Remove the constant of integration.

$e^{- x^{2}}$

Step 2

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

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Step 2.1

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

$e^{- x^{2}} \frac{d y}{d x} + e^{- x^{2}} (- 2 x y) = e^{- x^{2}} (6 x)$

Step 2.2

Rewrite using the commutative property of multiplication.

$e^{- x^{2}} \frac{d y}{d x} - 2 e^{- x^{2}} (x y) = e^{- x^{2}} (6 x)$

Step 2.3

Rewrite using the commutative property of multiplication.

$e^{- x^{2}} \frac{d y}{d x} - 2 e^{- x^{2}} (x y) = 6 e^{- x^{2}} x$

Step 2.4

Reorder factors in $e^{- x^{2}} \frac{d y}{d x} - 2 e^{- x^{2}} (x y) = 6 e^{- x^{2}} x$ .

$e^{- x^{2}} \frac{d y}{d x} - 2 x y e^{- x^{2}} = 6 x e^{- x^{2}}$

Step 3

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

$\frac{d}{d x} [e^{- x^{2}} y] = 6 x e^{- x^{2}}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{- x^{2}} y] d x = \int 6 x e^{- x^{2}} d x$

Step 5

Integrate the left side.

$e^{- x^{2}} y = \int 6 x e^{- x^{2}} d x$

Step 6

Integrate the right side.

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Step 6.1

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$e^{- x^{2}} y = 6 \int x e^{- x^{2}} d x$

Step 6.2

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

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Step 6.2.1

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

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Step 6.2.1.1

Differentiate $e^{- x^{2}}$ .

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

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 $g (x) = - x^{2}$ .

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Step 6.2.1.2.1

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

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

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 $a$ = $e$ .

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

Step 6.2.1.2.3

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

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

Step 6.2.1.3

Differentiate.

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Step 6.2.1.3.1

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

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

Step 6.2.1.3.2

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

$e^{- x^{2}} (- (2 x))$

Step 6.2.1.3.3

Multiply $2$ by $- 1$ .

$e^{- x^{2}} (- 2 x)$

Step 6.2.1.4

Simplify.

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Step 6.2.1.4.1

Reorder the factors of $e^{- x^{2}} (- 2 x)$ .

$- 2 e^{- x^{2}} x$

Step 6.2.1.4.2

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

$- 2 x e^{- x^{2}}$

Step 6.2.2

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

$e^{- x^{2}} y = 6 \int \frac{1}{- 2} d u_{2}$

Step 6.3

Move the negative in front of the fraction.

$e^{- x^{2}} y = 6 \int - \frac{1}{2} d u_{2}$

Step 6.4

Apply the constant rule.

$e^{- x^{2}} y = 6 (- \frac{1}{2} u_{2} + C)$

Step 6.5

Simplify the answer.

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Step 6.5.1

Simplify.

$e^{- x^{2}} y = 6 (- \frac{1}{2} u_{2}) + C$

Step 6.5.2

Simplify.

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Step 6.5.2.1

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

$e^{- x^{2}} y = 6 (- \frac{u_{2}}{2}) + C$

Step 6.5.2.2

Multiply $- 1$ by $6$ .

$e^{- x^{2}} y = - 6 \frac{u_{2}}{2} + C$

Step 6.5.2.3

Combine $- 6$ and $\frac{u_{2}}{2}$ .

$e^{- x^{2}} y = \frac{- 6 u_{2}}{2} + C$

Step 6.5.2.4

Cancel the common factor of $- 6$ and $2$ .

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Step 6.5.2.4.1

Factor $2$ out of $- 6 u_{2}$ .

$e^{- x^{2}} y = \frac{2 (- 3 u_{2})}{2} + C$

Step 6.5.2.4.2

Cancel the common factors.

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Step 6.5.2.4.2.1

Factor $2$ out of $2$ .

$e^{- x^{2}} y = \frac{2 (- 3 u_{2})}{2 (1)} + C$

Step 6.5.2.4.2.2

Cancel the common factor.

$e^{- x^{2}} y = \frac{2 (- 3 u_{2})}{2 \cdot 1} + C$

Step 6.5.2.4.2.3

Rewrite the expression.

$e^{- x^{2}} y = \frac{- 3 u_{2}}{1} + C$

Step 6.5.2.4.2.4

Divide $- 3 u_{2}$ by $1$ .

$e^{- x^{2}} y = - 3 u_{2} + C$

Step 6.5.3

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

$e^{- x^{2}} y = - 3 e^{- x^{2}} + C$

Step 7

Divide each term in $e^{- x^{2}} y = - 3 e^{- x^{2}} + C$ by $e^{- x^{2}}$ and simplify.

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Step 7.1

Divide each term in $e^{- x^{2}} y = - 3 e^{- x^{2}} + C$ by $e^{- x^{2}}$ .

$\frac{e^{- x^{2}} y}{e^{- x^{2}}} = \frac{- 3 e^{- x^{2}}}{e^{- x^{2}}} + \frac{C}{e^{- x^{2}}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

$\frac{e^{- x^{2}} y}{e^{- x^{2}}} = \frac{- 3 e^{- x^{2}}}{e^{- x^{2}}} + \frac{C}{e^{- x^{2}}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 3 e^{- x^{2}}}{e^{- x^{2}}} + \frac{C}{e^{- x^{2}}}$

Step 7.3

Simplify the right side.

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Step 7.3.1

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

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Step 7.3.1.1

Cancel the common factor.

$y = \frac{- 3 e^{- x^{2}}}{e^{- x^{2}}} + \frac{C}{e^{- x^{2}}}$

Step 7.3.1.2

Divide $- 3$ by $1$ .

$y = - 3 + \frac{C}{e^{- x^{2}}}$