Calculus Examples

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

Step 1

Subtract $x y$ from both sides of the equation.

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

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

Integrate $- x$ .

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Step 2.2.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

Remove the constant of integration.

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

Step 2.4

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

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

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

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

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

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Step 7.1.1

Let $u_{1} = - \frac{x^{2}}{2}$ . Find $\frac{d u_{1}}{d x}$ .

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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

Multiply $2$ by $- 1$ .

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

Step 7.1.1.4.2

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

$\frac{- 2}{2} x$

Step 7.1.1.4.3

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

$\frac{- 2 x}{2}$

Step 7.1.1.4.4

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

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Step 7.1.1.4.4.1

Factor $2$ out of $- 2 x$ .

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

Step 7.1.1.4.4.2

Cancel the common factors.

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Step 7.1.1.4.4.2.1

Factor $2$ out of $2$ .

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

Step 7.1.1.4.4.2.2

Cancel the common factor.

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

Step 7.1.1.4.4.2.3

Rewrite the expression.

$\frac{- x}{1}$

Step 7.1.1.4.4.2.4

Divide $- x$ by $1$ .

$- x$

Step 7.1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

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

Step 7.2

Since $- 1$ is constant with respect to $u_{1}$ , move $- 1$ out of the integral.

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

Step 7.3

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = e^{u_{1}}$ and $d v = \sqrt{- 2 u_{1}}$ .

$e^{- \frac{x^{2}}{2}} y = - (e^{u_{1}} (- \frac{1}{2} \cdot \frac{2}{3} {u_{2}}^{\frac{3}{2}}) - \int - \frac{1}{2} \cdot \frac{2}{3} {u_{2}}^{\frac{3}{2}} e^{u_{1}} d u_{1})$

Step 7.4

Simplify.

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Step 7.4.1

Multiply $\frac{2}{3}$ by $\frac{1}{2}$ .

$e^{- \frac{x^{2}}{2}} y = - (e^{u_{1}} (- \frac{2}{3 \cdot 2} {u_{2}}^{\frac{3}{2}}) - \int - \frac{1}{2} \cdot \frac{2}{3} {u_{2}}^{\frac{3}{2}} e^{u_{1}} d u_{1})$

Step 7.4.2

Multiply $3$ by $2$ .

$e^{- \frac{x^{2}}{2}} y = - (e^{u_{1}} (- \frac{2}{6} {u_{2}}^{\frac{3}{2}}) - \int - \frac{1}{2} \cdot \frac{2}{3} {u_{2}}^{\frac{3}{2}} e^{u_{1}} d u_{1})$

Step 7.4.3

Cancel the common factor of $2$ and $6$ .

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Step 7.4.3.1

Factor $2$ out of $2$ .

$e^{- \frac{x^{2}}{2}} y = - (e^{u_{1}} (- \frac{2 (1)}{6} {u_{2}}^{\frac{3}{2}}) - \int - \frac{1}{2} \cdot \frac{2}{3} {u_{2}}^{\frac{3}{2}} e^{u_{1}} d u_{1})$

Step 7.4.3.2

Cancel the common factors.

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Step 7.4.3.2.1

Factor $2$ out of $6$ .

$e^{- \frac{x^{2}}{2}} y = - (e^{u_{1}} (- \frac{2 \cdot 1}{2 \cdot 3} {u_{2}}^{\frac{3}{2}}) - \int - \frac{1}{2} \cdot \frac{2}{3} {u_{2}}^{\frac{3}{2}} e^{u_{1}} d u_{1})$

Step 7.4.3.2.2

Cancel the common factor.

$e^{- \frac{x^{2}}{2}} y = - (e^{u_{1}} (- \frac{2 \cdot 1}{2 \cdot 3} {u_{2}}^{\frac{3}{2}}) - \int - \frac{1}{2} \cdot \frac{2}{3} {u_{2}}^{\frac{3}{2}} e^{u_{1}} d u_{1})$

Step 7.4.3.2.3

Rewrite the expression.

$e^{- \frac{x^{2}}{2}} y = - (e^{u_{1}} (- \frac{1}{3} {u_{2}}^{\frac{3}{2}}) - \int - \frac{1}{2} \cdot \frac{2}{3} {u_{2}}^{\frac{3}{2}} e^{u_{1}} d u_{1})$

Step 7.4.4

Combine ${u_{2}}^{\frac{3}{2}}$ and $\frac{1}{3}$ .

$e^{- \frac{x^{2}}{2}} y = - (e^{u_{1}} (- \frac{{u_{2}}^{\frac{3}{2}}}{3}) - \int - \frac{1}{2} \cdot \frac{2}{3} {u_{2}}^{\frac{3}{2}} e^{u_{1}} d u_{1})$

Step 7.4.5

Combine $e^{u_{1}}$ and $\frac{{u_{2}}^{\frac{3}{2}}}{3}$ .

$e^{- \frac{x^{2}}{2}} y = - (- \frac{e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3} - \int - \frac{1}{2} \cdot \frac{2}{3} {u_{2}}^{\frac{3}{2}} e^{u_{1}} d u_{1})$

Step 7.4.6

Multiply $\frac{2}{3}$ by $\frac{1}{2}$ .

$e^{- \frac{x^{2}}{2}} y = - (- \frac{e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3} - \int - \frac{2}{3 \cdot 2} {u_{2}}^{\frac{3}{2}} e^{u_{1}} d u_{1})$

Step 7.4.7

Multiply $3$ by $2$ .

$e^{- \frac{x^{2}}{2}} y = - (- \frac{e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3} - \int - \frac{2}{6} {u_{2}}^{\frac{3}{2}} e^{u_{1}} d u_{1})$

Step 7.4.8

Cancel the common factor of $2$ and $6$ .

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Step 7.4.8.1

Factor $2$ out of $2$ .

$e^{- \frac{x^{2}}{2}} y = - (- \frac{e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3} - \int - \frac{2 (1)}{6} {u_{2}}^{\frac{3}{2}} e^{u_{1}} d u_{1})$

Step 7.4.8.2

Cancel the common factors.

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Step 7.4.8.2.1

Factor $2$ out of $6$ .

$e^{- \frac{x^{2}}{2}} y = - (- \frac{e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3} - \int - \frac{2 \cdot 1}{2 \cdot 3} {u_{2}}^{\frac{3}{2}} e^{u_{1}} d u_{1})$

Step 7.4.8.2.2

Cancel the common factor.

$e^{- \frac{x^{2}}{2}} y = - (- \frac{e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3} - \int - \frac{2 \cdot 1}{2 \cdot 3} {u_{2}}^{\frac{3}{2}} e^{u_{1}} d u_{1})$

Step 7.4.8.2.3

Rewrite the expression.

$e^{- \frac{x^{2}}{2}} y = - (- \frac{e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3} - \int - \frac{1}{3} {u_{2}}^{\frac{3}{2}} e^{u_{1}} d u_{1})$

Step 7.4.9

Combine ${u_{2}}^{\frac{3}{2}}$ and $\frac{1}{3}$ .

$e^{- \frac{x^{2}}{2}} y = - (- \frac{e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3} - \int - \frac{{u_{2}}^{\frac{3}{2}}}{3} e^{u_{1}} d u_{1})$

Step 7.4.10

Combine $e^{u_{1}}$ and $\frac{{u_{2}}^{\frac{3}{2}}}{3}$ .

$e^{- \frac{x^{2}}{2}} y = - (- \frac{e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3} - \int - \frac{e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3} d u_{1})$

Step 7.5

Since $- 1$ is constant with respect to $u_{1}$ , move $- 1$ out of the integral.

$e^{- \frac{x^{2}}{2}} y = - (- \frac{e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3} - - \int \frac{e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3} d u_{1})$

Step 7.6

Simplify.

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Step 7.6.1

Multiply $- 1$ by $- 1$ .

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

Step 7.6.2

Multiply $\int \frac{e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3} d u_{1}$ by $1$ .

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

Step 7.7

Since $\frac{{u_{2}}^{\frac{3}{2}}}{3}$ is constant with respect to $u_{1}$ , move $\frac{{u_{2}}^{\frac{3}{2}}}{3}$ out of the integral.

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

Step 7.8

The integral of $e^{u_{1}}$ with respect to $u_{1}$ is $e^{u_{1}}$ .

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

Step 7.9

Simplify.

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Step 7.9.1

Rewrite $- (- \frac{e^{u_{1}} {u_{2}}^{\frac{3}{2}}}{3} + \frac{{u_{2}}^{\frac{3}{2}}}{3} (e^{u_{1}} + C))$ as $- (- \frac{1}{3} e^{u_{1}} {u_{2}}^{\frac{3}{2}} + \frac{1}{3} {u_{2}}^{\frac{3}{2}} e^{u_{1}}) + C$ .

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

Step 7.9.2

Simplify.

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Step 7.9.2.1

Combine $e^{u_{1}}$ and $\frac{1}{3}$ .

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

Step 7.9.2.2

Combine ${u_{2}}^{\frac{3}{2}}$ and $\frac{e^{u_{1}}}{3}$ .

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

Step 7.9.2.3

Combine $\frac{1}{3}$ and ${u_{2}}^{\frac{3}{2}}$ .

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

Step 7.9.2.4

Combine $\frac{{u_{2}}^{\frac{3}{2}}}{3}$ and $e^{u_{1}}$ .

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

Step 7.9.2.5

Add $- \frac{{u_{2}}^{\frac{3}{2}} e^{u_{1}}}{3}$ and $\frac{{u_{2}}^{\frac{3}{2}} e^{u_{1}}}{3}$ .

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

Step 7.9.2.6

Multiply $- 1$ by $0$ .

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

Step 7.9.2.7

Add $0$ and $C$ .

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

Step 8

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

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Step 8.1

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

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

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.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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