Calculus Examples

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

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

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

Step 2.2

Rewrite using the commutative property of multiplication.

$e^{- x^{2}} \frac{d y}{d x} - 2 e^{- x^{2}} (x y) = e^{- x^{2}} (3 x^{2} e^{x^{2}})$

Step 2.3

Rewrite using the commutative property of multiplication.

$e^{- x^{2}} \frac{d y}{d x} - 2 e^{- x^{2}} (x y) = 3 e^{- x^{2}} (x^{2} e^{x^{2}})$

Step 2.4

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

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Step 2.4.1

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

$e^{- x^{2}} \frac{d y}{d x} - 2 e^{- x^{2}} (x y) = 3 (e^{x^{2}} e^{- x^{2}}) x^{2}$

Step 2.4.2

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

$e^{- x^{2}} \frac{d y}{d x} - 2 e^{- x^{2}} (x y) = 3 e^{x^{2} - x^{2}} x^{2}$

Step 2.4.3

Subtract $x^{2}$ from $x^{2}$ .

$e^{- x^{2}} \frac{d y}{d x} - 2 e^{- x^{2}} (x y) = 3 e^{0} x^{2}$

Step 2.5

Simplify $3 e^{0} x^{2}$ .

$e^{- x^{2}} \frac{d y}{d x} - 2 e^{- x^{2}} (x y) = 3 x^{2}$

Step 2.6

Reorder factors in $e^{- x^{2}} \frac{d y}{d x} - 2 e^{- x^{2}} (x y) = 3 x^{2}$ .

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

Step 3

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

$\frac{d}{d x} [e^{- x^{2}} y] = 3 x^{2}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{- x^{2}} y] d x = \int 3 x^{2} d x$

Step 5

Integrate the left side.

$e^{- x^{2}} y = \int 3 x^{2} d x$

Step 6

Integrate the right side.

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Step 6.1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$e^{- x^{2}} y = 3 \int x^{2} d x$

Step 6.2

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

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

Step 6.3

Simplify the answer.

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Step 6.3.1

Rewrite $3 (\frac{1}{3} x^{3} + C)$ as $3 (\frac{1}{3}) x^{3} + C$ .

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

Step 6.3.2

Simplify.

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Step 6.3.2.1

Combine $3$ and $\frac{1}{3}$ .

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

Step 6.3.2.2

Cancel the common factor of $3$ .

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Step 6.3.2.2.1

Cancel the common factor.

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

Step 6.3.2.2.2

Rewrite the expression.

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

Step 6.3.2.3

Multiply $x^{3}$ by $1$ .

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

Step 7

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

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Step 7.1

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

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

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.

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

Step 7.2.1.2

Divide $y$ by $1$ .

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