Calculus Examples

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

Step 1

Add $2 x e^{- x^{2}}$ to both sides of the equation.

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

Step 2

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

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Step 2.1

Set up the integration.

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

Step 2.2

Integrate $2 x$ .

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Step 2.2.1

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

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

Step 2.2.2

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 2.2.3

Simplify the answer.

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Step 2.2.3.1

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 2.2.3.2

Simplify.

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Step 2.2.3.2.1

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

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

Step 2.2.3.2.2

Cancel the common factor of $2$ .

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Step 2.2.3.2.2.1

Cancel the common factor.

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

Step 2.2.3.2.2.2

Rewrite the expression.

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

Step 2.2.3.2.3

Multiply $x^{2}$ by $1$ .

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

Step 2.3

Remove the constant of integration.

$e^{x^{2}}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

Multiply $e^{x^{2}}$ by $e^{- x^{2}}$ by adding the exponents.

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Step 3.4.1

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

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

Step 3.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.4.3

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

$e^{x^{2}} \frac{d y}{d x} + 2 e^{x^{2}} (x y) = 2 e^{0} x$

Step 3.5

Simplify $2 e^{0} x$ .

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

Step 3.6

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Simplify the answer.

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Step 7.3.1

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

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

Step 7.3.2

Simplify.

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Step 7.3.2.1

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

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

Step 7.3.2.2

Cancel the common factor of $2$ .

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Step 7.3.2.2.1

Cancel the common factor.

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

Step 7.3.2.2.2

Rewrite the expression.

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

Step 7.3.2.3

Multiply $x^{2}$ by $1$ .

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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