Calculus Examples

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

Step 1

Add $x y$ to both sides of the equation.

$\frac{d y}{d x} + x y = x^{3}$

Step 2

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

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Step 2.1

Set up the integration.

$e^{\int x d x}$

Step 2.2

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 2.3

Remove the constant of integration.

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

Step 2.4

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

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

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Step 7.1.1

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

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Step 7.1.1.1

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

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

Step 7.1.1.2

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

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

Step 7.1.1.3

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

$\frac{1}{2} (2 x)$

Step 7.1.1.4

Simplify terms.

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Step 7.1.1.4.1

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

$\frac{2}{2} x$

Step 7.1.1.4.2

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

$\frac{2 x}{2}$

Step 7.1.1.4.3

Cancel the common factor of $2$ .

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Step 7.1.1.4.3.1

Cancel the common factor.

$\frac{2 x}{2}$

Step 7.1.1.4.3.2

Divide $x$ by $1$ .

$x$

Step 7.1.2

Rewrite the problem using $u$ and $d u$ .

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

Step 7.2

Rewrite ${\sqrt{2 u}}^{2}$ as $2 u$ .

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Step 7.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 u}$ as ${(2 u)}^{\frac{1}{2}}$ .

$e^{\frac{x^{2}}{2}} y = \int {({(2 u)}^{\frac{1}{2}})}^{2} e^{u} d u$

Step 7.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$e^{\frac{x^{2}}{2}} y = \int {(2 u)}^{\frac{1}{2} \cdot 2} e^{u} d u$

Step 7.2.3

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

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

Step 7.2.4

Cancel the common factor of $2$ .

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Step 7.2.4.1

Cancel the common factor.

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

Step 7.2.4.2

Rewrite the expression.

$e^{\frac{x^{2}}{2}} y = \int {(2 u)}^{1} e^{u} d u$

Step 7.2.5

Simplify.

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

Step 7.3

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

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

Step 7.4

Integrate by parts using the formula $\int w d v = w v - \int v d w$ , where $w = u$ and $dv = e^{u}$ .

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

Step 7.5

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

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

Step 7.6

Simplify.

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

Step 7.7

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

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

Step 7.8

Simplify.

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Step 7.8.1

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

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

Step 7.8.2

Apply the distributive property.

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

Step 7.8.3

Cancel the common factor of $2$ .

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Step 7.8.3.1

Cancel the common factor.

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

Step 7.8.3.2

Rewrite the expression.

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

Step 7.8.4

Multiply $- 1$ by $2$ .

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

Step 7.9

Reorder terms.

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

Cancel the common factor of $e^{\frac{1}{2} x^{2}}$ and $e^{\frac{x^{2}}{2}}$ .

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Step 8.3.1.1.1

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

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

Step 8.3.1.1.2

Cancel the common factors.

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Step 8.3.1.1.2.1

Multiply by $1$ .

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

Step 8.3.1.1.2.2

Cancel the common factor.

Step 8.3.1.1.2.3

Rewrite the expression.

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

Step 8.3.1.1.2.4

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

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

Step 8.3.1.2

Simplify the numerator.

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Step 8.3.1.2.1

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

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Step 8.3.1.2.1.1

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

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

Step 8.3.1.2.1.2

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

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

Step 8.3.1.2.1.3

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

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

Step 8.3.1.2.2

Cancel the common factor of $2$ .

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Step 8.3.1.2.2.1

Cancel the common factor.

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

Step 8.3.1.2.2.2

Rewrite the expression.

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

Step 8.3.1.2.3

Subtract $1$ from $1$ .

$y = x^{2} e^{\frac{x^{2} \cdot 0}{2}} + \frac{- 2 e^{\frac{1}{2} x^{2}}}{e^{\frac{x^{2}}{2}}} + \frac{C}{e^{\frac{x^{2}}{2}}}$

Step 8.3.1.3

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

$y = x^{2} e^{\frac{0}{2}} + \frac{- 2 e^{\frac{1}{2} x^{2}}}{e^{\frac{x^{2}}{2}}} + \frac{C}{e^{\frac{x^{2}}{2}}}$

Step 8.3.1.4

Divide $0$ by $2$ .

$y = x^{2} e^{0} + \frac{- 2 e^{\frac{1}{2} x^{2}}}{e^{\frac{x^{2}}{2}}} + \frac{C}{e^{\frac{x^{2}}{2}}}$

Step 8.3.1.5

Anything raised to $0$ is $1$ .

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

Step 8.3.1.6

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

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

Step 8.3.1.7

Cancel the common factor of $e^{\frac{1}{2} x^{2}}$ and $e^{\frac{x^{2}}{2}}$ .

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Step 8.3.1.7.1

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

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

Step 8.3.1.7.2

Cancel the common factors.

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Step 8.3.1.7.2.1

Multiply by $1$ .

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

Step 8.3.1.7.2.2

Cancel the common factor.

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

Step 8.3.1.7.2.3

Rewrite the expression.

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

Step 8.3.1.7.2.4

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

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

Step 8.3.1.8

Simplify the numerator.

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Step 8.3.1.8.1

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

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Step 8.3.1.8.1.1

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

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

Step 8.3.1.8.1.2

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

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

Step 8.3.1.8.1.3

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

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

Step 8.3.1.8.2

Cancel the common factor of $2$ .

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Step 8.3.1.8.2.1

Cancel the common factor.

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

Step 8.3.1.8.2.2

Rewrite the expression.

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

Step 8.3.1.8.3

Subtract $1$ from $1$ .

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

Step 8.3.1.9

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

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

Step 8.3.1.10

Divide $0$ by $2$ .

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

Step 8.3.1.11

Anything raised to $0$ is $1$ .

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

Step 8.3.1.12

Multiply $- 2$ by $1$ .

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