Calculus Examples

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

Step 1

Add $2 x y$ to both sides of the equation.

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

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

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

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

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Step 7.1.1

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

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Step 7.1.1.1

Differentiate $x^{2}$ .

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

Step 7.1.1.2

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

$2 x$

Step 7.1.2

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

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

Step 7.2

Simplify.

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 7.4

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

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

Step 7.5

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

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

Step 7.6

Simplify.

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

Step 7.7

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

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

Step 7.8

Simplify.

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Step 7.8.1

Apply the distributive property.

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

Step 7.8.2

Multiply $\frac{1}{2} (x^{2} e^{x^{2}})$ .

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Step 7.8.2.1

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

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

Step 7.8.2.2

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

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

Step 7.8.3

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

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

Step 7.9

Reorder terms.

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

Step 8

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

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Step 8.1

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

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

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.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

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

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Step 8.3.1.1.1

Cancel the common factor.

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

Step 8.3.1.1.2

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

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

Step 8.3.1.2

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

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Step 8.3.1.2.1

Cancel the common factor.

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

Step 8.3.1.2.2

Divide $- \frac{1}{2}$ by $1$ .

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

Step 8.3.2

Subtract $\frac{1}{2}$ from $\frac{1}{2} x^{2}$ .

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Step 8.3.2.1

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

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

Step 8.3.2.2

To write $x^{2} \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 8.3.2.3

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

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

Step 8.3.2.4

Combine the numerators over the common denominator.

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

Step 8.3.3

Simplify the numerator.

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Step 8.3.3.1

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

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

Step 8.3.3.2

Cancel the common factor of $2$ .

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Step 8.3.3.2.1

Cancel the common factor.

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

Step 8.3.3.2.2

Rewrite the expression.

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

Step 8.3.3.3

Rewrite $x^{2} - 1$ in a factored form.

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Step 8.3.3.3.1

Rewrite $1$ as $1^{2}$ .

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

Step 8.3.3.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 8.3.4

To write $\frac{(x + 1) (x - 1)}{2}$ as a fraction with a common denominator, multiply by $\frac{e^{x^{2}}}{e^{x^{2}}}$ .

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

Step 8.3.5

To write $\frac{C}{e^{x^{2}}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 8.3.6

Write each expression with a common denominator of $2 e^{x^{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.3.6.1

Multiply $\frac{(x + 1) (x - 1)}{2}$ by $\frac{e^{x^{2}}}{e^{x^{2}}}$ .

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

Step 8.3.6.2

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

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

Step 8.3.6.3

Reorder the factors of $e^{x^{2}} \cdot 2$ .

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

Step 8.3.7

Combine the numerators over the common denominator.

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

Step 8.3.8

Simplify the numerator.

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Step 8.3.8.1

Expand $(x + 1) (x - 1)$ using the FOIL Method.

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Step 8.3.8.1.1

Apply the distributive property.

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

Step 8.3.8.1.2

Apply the distributive property.

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

Step 8.3.8.1.3

Apply the distributive property.

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

Step 8.3.8.2

Simplify and combine like terms.

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Step 8.3.8.2.1

Simplify each term.

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Step 8.3.8.2.1.1

Multiply $x$ by $x$ .

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

Step 8.3.8.2.1.2

Move $- 1$ to the left of $x$ .

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

Step 8.3.8.2.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 8.3.8.2.1.4

Multiply $x$ by $1$ .

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

Step 8.3.8.2.1.5

Multiply $- 1$ by $1$ .

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

Step 8.3.8.2.2

Add $- x$ and $x$ .

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

Step 8.3.8.2.3

Add $x^{2}$ and $0$ .

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

Step 8.3.8.3

Apply the distributive property.

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

Step 8.3.8.4

Rewrite $- 1 e^{x^{2}}$ as $- e^{x^{2}}$ .

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

Step 8.3.8.5

Move $2$ to the left of $C$ .

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