Calculus Examples

$\frac{d y}{d x} + 2 x y = 16 x$

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 $2$ .

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Step 1.2.3.2.2.1

Cancel the common factor.

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

Step 1.2.3.2.2.2

Rewrite the expression.

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

Step 1.2.3.2.3

Multiply $x^{2}$ 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 $e^{x^{2}}$ .

$e^{x^{2}} \frac{d y}{d x} + e^{x^{2}} (2 x y) = e^{x^{2}} (16 x)$

Step 2.2

Rewrite using the commutative property of multiplication.

$e^{x^{2}} \frac{d y}{d x} + 2 e^{x^{2}} (x y) = e^{x^{2}} (16 x)$

Step 2.3

Rewrite using the commutative property of multiplication.

$e^{x^{2}} \frac{d y}{d x} + 2 e^{x^{2}} (x y) = 16 e^{x^{2}} x$

Step 2.4

Reorder factors in $e^{x^{2}} \frac{d y}{d x} + 2 e^{x^{2}} (x y) = 16 e^{x^{2}} x$ .

$e^{x^{2}} \frac{d y}{d x} + 2 x y e^{x^{2}} = 16 x e^{x^{2}}$

Step 3

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

$\frac{d}{d x} [e^{x^{2}} y] = 16 x e^{x^{2}}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{x^{2}} y] d x = \int 16 x e^{x^{2}} d x$

Step 5

Integrate the left side.

$e^{x^{2}} y = \int 16 x e^{x^{2}} d x$

Step 6

Integrate the right side.

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Step 6.1

Since $16$ is constant with respect to $x$ , move $16$ out of the integral.

$e^{x^{2}} y = 16 \int x e^{x^{2}} d x$

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 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 6.2.1.4

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 $d u_{2}$ .

$e^{x^{2}} y = 16 \int \frac{1}{2} d u_{2}$

Step 6.3

Apply the constant rule.

$e^{x^{2}} y = 16 (\frac{1}{2} u_{2} + C)$

Step 6.4

Simplify the answer.

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Step 6.4.1

Rewrite $16 (\frac{1}{2} u_{2} + C)$ as $16 (\frac{1}{2}) u_{2} + C$ .

$e^{x^{2}} y = 16 (\frac{1}{2}) u_{2} + C$

Step 6.4.2

Simplify.

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Step 6.4.2.1

Combine $16$ and $\frac{1}{2}$ .

$e^{x^{2}} y = \frac{16}{2} u_{2} + C$

Step 6.4.2.2

Cancel the common factor of $16$ and $2$ .

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Step 6.4.2.2.1

Factor $2$ out of $16$ .

$e^{x^{2}} y = \frac{2 \cdot 8}{2} u_{2} + C$

Step 6.4.2.2.2

Cancel the common factors.

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Step 6.4.2.2.2.1

Factor $2$ out of $2$ .

$e^{x^{2}} y = \frac{2 \cdot 8}{2 (1)} u_{2} + C$

Step 6.4.2.2.2.2

Cancel the common factor.

$e^{x^{2}} y = \frac{2 \cdot 8}{2 \cdot 1} u_{2} + C$

Step 6.4.2.2.2.3

Rewrite the expression.

$e^{x^{2}} y = \frac{8}{1} u_{2} + C$

Step 6.4.2.2.2.4

Divide $8$ by $1$ .

$e^{x^{2}} y = 8 u_{2} + C$

Step 6.4.3

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

$e^{x^{2}} y = 8 e^{x^{2}} + C$

Step 7

Divide each term in $e^{x^{2}} y = 8 e^{x^{2}} + C$ by $e^{x^{2}}$ and simplify.

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Step 7.1

Divide each term in $e^{x^{2}} y = 8 e^{x^{2}} + C$ by $e^{x^{2}}$ .

$\frac{e^{x^{2}} y}{e^{x^{2}}} = \frac{8 e^{x^{2}}}{e^{x^{2}}} + \frac{C}{e^{x^{2}}}$

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.

$\frac{e^{x^{2}} y}{e^{x^{2}}} = \frac{8 e^{x^{2}}}{e^{x^{2}}} + \frac{C}{e^{x^{2}}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{8 e^{x^{2}}}{e^{x^{2}}} + \frac{C}{e^{x^{2}}}$

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.

$y = \frac{8 e^{x^{2}}}{e^{x^{2}}} + \frac{C}{e^{x^{2}}}$

Step 7.3.1.2

Divide $8$ by $1$ .

$y = 8 + \frac{C}{e^{x^{2}}}$